{"version":1,"lectureId":"01M1QM8AGRW010421VMGE316AV","attempt":0,"publication":{"slug":"understanding-the-quadratic-formula-and-how-it-works","title":"Understanding the Quadratic Formula and How It Works","subject":"mathematics","summary":"The quadratic formula is not a rule to memorise: it is one algebraic move, completing the square, carried out once with letters instead of numbers. This lecture starts from a parabola and what a root actually is, shows why factoring runs out, then completes the square on a concrete equation with the square literally drawn as a square and one corner missing. The same five steps are then repeated with a, b and c, producing the formula line by line, and the result is tested against the parabola we began with. The last part reads the discriminant, b squared minus four a c, off a moving picture: two crossings, then one, then none, and the complex conjugate pair that the algebra returns when the parabola misses the axis.","metaDescription":"Derive the quadratic formula by completing the square, then read the discriminant to see two real roots, one, or a complex pair.","transcript":"Here is a question that covers every quadratic equation there is. Solve a x squared plus b x plus c equals zero, for x, whatever the three numbers a, b and c happen to be. By the end we will have one line of algebra that answers it every single time. A quadratic is easiest to meet as a picture. This blue curve is y equals x squared minus x minus six. One bend, opening upward, and that shape is called a parabola. Solving the equation means finding the x values where y is zero, and a height of zero means the curve is sitting on the horizontal axis. This one meets the axis in two places, here and here. Those two places are at minus two and at three. Those numbers are the roots of the equation, and they are what the whole lecture is about. For this particular one we can find them with no machinery at all. The left-hand side factors: x squared minus x minus six is x minus three, times x plus two. And a product can only be zero when one of the two factors is zero. So x equals three, or x equals minus two, which is exactly what the picture already told us. Now change the three numbers and try again. This green curve is y equals x squared minus four x plus one, and it crosses the axis twice as well. But try factoring this one. Nothing with whole numbers works, and the reason is visible on the picture: the crossings are not whole numbers. They are two minus the square root of three, and two plus the square root of three. Guessing factors is a trick that works when it works. What we want is something that never fails, and here it is: the quadratic formula. Read it once. Minus b at the front. Then plus or minus a square root. Under that root, b squared minus four a c. And the whole thing divided by two a. That piece under the root has a name of its own: the discriminant. Whether a parabola crosses the axis twice, touches it once, or misses it entirely is decided by that one expression, and the last part of the lecture is about exactly that. Everything rests on one small fact: a square is easy to undo. If something squared is sixteen, then that something is four, or minus four. So the plan is to force any quadratic into exactly that shape. Here is a concrete one to practise on. x squared plus six x minus seven equals zero. It does happen to factor, but ignore that and watch a different route. Move the seven out of the way first, so that the two x terms stand alone. x squared plus six x equals seven. Now draw x squared as an actual square, with side x. And draw six x as two rectangles, three wide and x tall, laid against two of its edges. Look at the shape those three pieces make. It is a big square with one corner missing, and the missing corner is three by three. Its area is nine. So add nine to the left-hand side. And add nine to the right as well, to keep the equation true. That gives x squared plus six x plus nine equals sixteen. And the left-hand side is now a completed square. The picture says the same thing: the whole edge along the bottom is x plus three, so the area is x plus three, all squared, and that area is sixteen. Before finishing the algebra, put the parabola on the screen. y equals x squared plus six x minus seven. Its two crossings are the answers we are about to compute. Take the square root of both sides. Sixteen has two square roots, four and minus four, so plus or minus arrives right here, and this is the only place in the whole method that it comes from. Then subtract three from both sides. Minus three plus four is one, and minus three minus four is minus seven. There they are on the curve. And notice where those two roots sit. The parabola turns at x equals minus three, and the roots are four units to the left of that and four units to the right. The plus or minus is the symmetry of the curve, written as one symbol. That is the entire trick, and nothing in it cared that the numbers were six and seven. So let us run it again with letters. Same five moves, with a, b and c standing in for whatever the numbers are. Start from the general equation, with a not equal to zero, because otherwise there is no x squared term and no quadratic. The trick needs the x squared term to have nothing in front of it, so divide every single term by a. That is allowed, precisely because a is not zero. Move the constant across to the right, exactly as we moved the seven before. Now the key step. Last time we added three squared, and three was half of six, the coefficient of x. Here the coefficient of x is b over a, half of that is b over two a, and so what we add to both sides is b over two a, all squared. The left-hand side is now a completed square, and the number inside the bracket is that same half: x plus b over two a, all squared. On the right, b over two a squared is b squared over four a squared. Put the right-hand side over a single denominator. Multiply c over a top and bottom by four a, and the two fractions combine into one. Look at that numerator. b squared minus four a c. Nobody put it there; it fell out of the arithmetic, and it is the expression we will read the geometry off in the last part. Take the square root of both sides. The square root of four a squared is two a, and a square root comes with two signs, so plus or minus appears here, once, and never again. Then subtract b over two a from both sides, and x stands alone. Both of those terms already have the same denominator, two a, so write them as a single fraction. And that is the quadratic formula. There it is on its own. Minus b, plus or minus the square root of b squared minus four a c, all divided by two a. Nothing in that line was guessed. One warning before we test it. The letters have to come from an equation with zero on the right. If yours is not in that shape yet, put it in that shape first, and keep every minus sign as you read the numbers off. Test it on the very first parabola. y equals x squared minus x minus six, so a is one, b is minus one, and c is minus six. Substitute. Minus b is plus one. b squared is one, and four a c is minus twenty four, so under the root we have one plus twenty four. One plus twenty four is twenty five. And the square root of twenty five is exactly five, so the awkward part disappears. So x is one plus five over two, or one minus five over two. Three, or minus two. The formula lands on the same two crossings the picture showed us at the start. It worked here, and it will work where nothing factors, because we never once assumed the numbers were friendly. What is left is to read the formula properly, and everything interesting is hiding under that square root. Here is one parabola that I can slide up and down. Its equation is x squared minus two x plus c, and c is the only number I am going to touch. At c equals zero it crosses the axis in two places, at zero and at two. The dashed line is the axis of symmetry, halfway between them, at x equals one. For this family a is one and b is minus two, so b squared minus four a c is four minus four c. At c equals zero that reads four, a positive number, and I will keep it on the screen while c moves. Now lift the parabola. As c grows, that number shrinks, and the two crossings walk in toward the dashed line together. Keep going. At c equals one it is exactly zero, and the two crossings arrive at the same place. The parabola touches the axis there and does not pass through it. One root, or a repeated root, depending on how you like to count. And if I lifted it any further there would be nothing left on the axis to mark, because four minus four c would be negative. Let me put all three possibilities beside each other. Three parabolas of the same shape, three values of c. On the left the discriminant is positive. Two crossings, and two real roots. In the middle it is zero. The turning point of the parabola sits exactly on the axis, and the two roots have collapsed into one. On the right it is negative. The parabola clears the axis completely, so there is no real number at all that you could put in for x and get zero out. But no real root is not the same as no root. Take that third case on its own. x squared minus two x plus two equals zero, so a is one, b is minus two, and c is two. Under the root we get four minus eight, which is minus four. No real number squares to something negative, so this is where we widen what counts as a number: the square root of minus four is two i, where i squared is minus one. Divide the two through, and the roots are one plus i and one minus i. A conjugate pair, and look where they sit: either side of x equals one, which is the axis of symmetry, exactly as the real pairs did. So the same formula answers all three cases, and it always answers them in the same shape: a centre, and a distance either side of it. Split the formula that way and it reads out loud. Minus b over two a is the centre, the axis of symmetry. The square root of the discriminant over two a is how far out the roots lie. Three things to carry away. The formula is nothing more than completing the square, done once with letters instead of numbers. Minus b over two a is the centre of the picture, and the two roots are always symmetric about it. And b squared minus four a c decides the whole story. Positive, two real roots. Zero, one. Negative, a conjugate pair, and a parabola that never touches the axis at all.","watch":{"version":1,"scenes":[{"title":"What a Root Is","start":0,"end":142.6826875,"objects":{"answers":"a Math [text] that says \"$x = 3 quad upright(\"or\") quad x = -2$\"","axes":"an Axes (x_range=(-4.0, 5.0), y_range=(-8.0, 8.0), x_ticks_every=1.0)","curve":"a FunctionPlot [blue] labelled \"y = x^2 - x - 6\" drawn in axes (function=<function>, x_range=(-3.2, 4.2))","factored":"a Math [text] that says \"$x^2 - x - 6 = (x - 3)(x + 2)$\"","formula":"a Math [text] that says \"$x = frac(- b plus.minus sqrt(b^2 - 4 a c), 2 a)$\"","hard_a":"a PlotPoint [red] labelled \"2 - sqrt(3)\" drawn in axes (target='hard_curve', x=0.2679491924311228)","hard_b":"a PlotPoint [red] labelled \"2 + sqrt(3)\" drawn in axes (target='hard_curve', x=3.732050807568877)","hard_curve":"a FunctionPlot [green] labelled \"y = x^2 - 4 x + 1\" drawn in axes (function=<function>, x_range=(-0.6, 4.6))","hard_eq":"a Math [text] that says \"$x^2 - 4 x + 1 = 0$\"","question":"a Panel that says \"Solve $a x^2 + b x + c = 0$ for $x$, for every choice of $a$, $b$ and $c$, with $a eq.not 0$.\"","root_a":"a PlotPoint [red] labelled \"-2\" drawn in axes (target='curve', x=-2.0)","root_b":"a PlotPoint [red] labelled \"3\" drawn in axes (target='curve', x=3.0)"},"beats":[{"start":0,"say":"Here is a question that covers every quadratic equation there is. 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The plus or minus is the symmetry of the curve, written as one symbol.","live":["eq3","eq4","eq5","eq6","axes2","heading2","curve3","r1","r2"],"does":[[268.4916875,"vsym is shown on the screen, written out."],[272.4396875,"The segment (-3.0, 0.0) to (-7.0, 0.0) in axes2 is lit up."],[274.3316875,"The segment (-3.0, 0.0) to (1.0, 0.0) in axes2 is lit up."],[277.0256875,"eq4 (the \"plus.minus\" part) is emphasized."],[280.19568749999996,"axes2: retire a lit segment (unemphasize_line)."],[280.19568749999996,"axes2: retire a lit segment (unemphasize_line)."],[280.19568749999996,"eq4 (the \"plus.minus\" part) is no longer emphasized."]]},{"start":280.7956875,"say":"That is the entire trick, and nothing in it cared that the numbers were six and seven. So let us run it again with letters.","live":["eq3","eq4","eq5","eq6","axes2","heading2","curve3","r1","r2","vsym"],"does":[[288.6204375,"axes2 is hidden from the screen — left the board."],[288.6204375,"curve3 is hidden from the screen — axes2 left the board."],[288.6204375,"r1 is hidden from the screen — axes2 left the board."],[288.6204375,"r2 is hidden from the screen — axes2 left the board."],[288.6204375,"vsym is hidden from the screen — axes2 left the board."],[288.6204375,"eq3 is hidden from the screen — left the board."],[288.6204375,"eq4 is hidden from the screen — left the board."],[288.6204375,"eq5 is hidden from the screen — left the board."],[288.6204375,"eq6 is hidden from the screen — left the board."],[288.6204375,"heading2 is hidden from the screen — left the board."]]}]},{"title":"The Same Steps, With Letters","start":289.66210416666667,"end":480.45827083333336,"objects":{"axes3":"an Axes (x_range=(-4.0, 5.0), y_range=(-8.0, 8.0), x_ticks_every=1.0)","c_left":"a PlotPoint [red] labelled \"-2\" drawn in axes3 (target='curve5', x=-2.0)","c_right":"a PlotPoint [red] labelled \"3\" drawn in axes3 (target='curve5', x=3.0)","curve5":"a FunctionPlot [blue] labelled \"y = x^2 - x - 6\" drawn in axes3 (function=<function>, x_range=(-3.2, 4.2))","heading_a":"a Heading that says \"Completing the Square in General\"","heading_b":"a Heading that says \"Undoing the Square\"","heading_c":"a Heading that says \"The Quadratic Formula\"","heading_d":"a Heading that says \"Does It Work?\"","l1":"a Math [text] that says \"$a x^2 + b x + c = 0$\"","l2":"a Math [text] that says \"$x^2 + frac(b, a) x + frac(c, a) = 0$\"","l3":"a Math [text] that says \"$x^2 + frac(b, a) x = - frac(c, a)$\"","l4":"a Math [text] that says \"$x^2 + frac(b, a) x + (frac(b, 2 a))^2 = - frac(c, a) + (frac(b, 2 a))^2$\"","l5":"a Math [text] that says \"$(x + frac(b, 2 a))^2 = frac(b^2, 4 a^2) - frac(c, a)$\"","l6":"a Math [text] that says \"$(x + frac(b, 2 a))^2 = frac(b^2 - 4 a c, 4 a^2)$\"","l7":"a Math [text] that says \"$x + frac(b, 2 a) = plus.minus frac(sqrt(b^2 - 4 a c), 2 a)$\"","l8":"a Math [text] that says \"$x = - frac(b, 2 a) plus.minus frac(sqrt(b^2 - 4 a c), 2 a)$\"","l9":"a Math [text] that says \"$x = frac(- b plus.minus sqrt(b^2 - 4 a c), 2 a)$\"","note_def":"a Panel that says \"First rearrange until there is zero on the right. Only then read $a$, $b$ and $c$ off the equation, signs included, and substitute.\"","sub0":"a Math [text] that says \"$a = 1, quad b = -1, quad c = -6$\"","sub1":"a Math [text] that says \"$x = frac(1 plus.minus sqrt(1 + 24), 2)$\"","sub2":"a Math [text] that says \"$x = 3 quad upright(\"or\") quad x = -2$\""},"beats":[{"start":289.66210416666667,"say":"Same five moves, with a, b and c standing in for whatever the numbers are. Start from the general equation, with a not equal to zero, because otherwise there is no x squared term and no quadratic.","live":[],"does":[[289.66210416666667,"heading_a is shown on the screen, written out."],[294.75910416666665,"l1 is shown on the screen, written out."]]},{"start":302.3366041666667,"say":"The trick needs the x squared term to have nothing in front of it, so divide every single term by a. That is allowed, precisely because a is not zero.","live":["l1","heading_a"],"does":[[306.17910416666666,"l2 is shown on the screen, written out."]]},{"start":312.42160416666667,"say":"Move the constant across to the right, exactly as we moved the seven before.","live":["l1","l2","heading_a"],"does":[[312.77010416666667,"l3 is shown on the screen, written out."]]},{"start":318.01410416666664,"say":"Now the key step. Last time we added three squared, and three was half of six, the coefficient of x. Here the coefficient of x is b over a, half of that is b over two a, and so what we add to both sides is b over two a, all squared.","live":["l1","l2","l3","heading_a"],"does":[[331.38910416666664,"l4 is shown on the screen, written out."],[331.7841041666667,"l4 (the \"(frac(b, 2 a))^2#2\" part) is indicated — a transient flash."],[334.84860416666663,"l4 moves to a new place on the board."],[334.84860416666663,"l1 is hidden from the screen — left the board."],[334.84860416666663,"l2 is hidden from the screen — left the board."],[334.84860416666663,"l3 is hidden from the screen — left the board."]]},{"start":335.44860416666666,"say":"The left-hand side is now a completed square, and the number inside the bracket is that same half: x plus b over two a, all squared. On the right, b over two a squared is b squared over four a squared.","live":["l4","heading_a"],"does":[[337.28310416666665,"l5 is shown on the screen, written out."]]},{"start":350.07410416666664,"say":"Put the right-hand side over a single denominator. Multiply c over a top and bottom by four a, and the two fractions combine into one.","live":["l4","heading_a","l5"],"does":[[357.80610416666667,"l6 is shown on the screen, written out."]]},{"start":359.8221041666667,"say":"Look at that numerator. b squared minus four a c. Nobody put it there; it fell out of the arithmetic, and it is the expression we will read the geometry off in the last part.","live":["l4","heading_a","l5","l6"],"does":[[360.68110416666667,"l6 (the \"b^2 - 4 a c\" part) is emphasized."],[370.4571041666667,"l6 (the \"b^2 - 4 a c\" part) is no longer emphasized."],[371.51360416666665,"l6 moves to a new place on the board."],[371.51360416666665,"heading_a is hidden from the screen — left the board."],[371.51360416666665,"l4 is hidden from the screen — left the board."],[371.51360416666665,"l5 is hidden from the screen — left the board."]]},{"start":372.1136041666667,"say":"Take the square root of both sides. The square root of four a squared is two a, and a square root comes with two signs, so plus or minus appears here, once, and never again.","live":["l6"],"does":[[372.1136041666667,"heading_b is shown on the screen, written out."],[373.18110416666667,"l7 is shown on the screen, written out."],[378.7891041666667,"l7 (the \"plus.minus\" part) is emphasized."],[382.35310416666664,"l7 (the \"plus.minus\" part) is no longer emphasized."]]},{"start":384.03310416666665,"say":"Then subtract b over two a from both sides, and x stands alone.","live":["l6","l7","heading_b"],"does":[[384.6951041666666,"l8 is shown on the screen, written out."]]},{"start":389.85710416666666,"say":"Both of those terms already have the same denominator, two a, so write them as a single fraction. And that is the quadratic formula.","live":["l6","l7","l8","heading_b"],"does":[[394.5821041666667,"l9 is shown on the screen, written out."],[398.19860416666666,"l9 moves to a new place on the board."],[398.19860416666666,"heading_b is hidden from the screen — left the board."],[398.19860416666666,"l6 is hidden from the screen — left the board."],[398.19860416666666,"l7 is hidden from the screen — left the board."],[398.19860416666666,"l8 is hidden from the screen — left the board."]]},{"start":398.7986041666667,"say":"There it is on its own. Minus b, plus or minus the square root of b squared minus four a c, all divided by two a. Nothing in that line was guessed.","live":["l9"],"does":[[398.7986041666667,"heading_c is shown on the screen, written out."],[409.40510416666666,"A box is drawn around l9."]]},{"start":410.84060416666665,"say":"One warning before we test it. The letters have to come from an equation with zero on the right. If yours is not in that shape yet, put it in that shape first, and keep every minus sign as you read the numbers off.","live":["l9","heading_c"],"does":[[411.46810416666665,"note_def is shown on the screen, written out."],[415.35710416666666,"note_def (the \"zero on the right\" part) is emphasized."],[419.8611041666666,"note_def (the \"zero on the right\" part) is no longer emphasized."],[423.6926041666667,"l9 moves to a new place on the board."],[423.6926041666667,"heading_c is hidden from the screen — left the board."],[423.6926041666667,"note_def is hidden from the screen — left the board."]]},{"start":424.8926041666667,"say":"Test it on the very first parabola. y equals x squared minus x minus six, so a is one, b is minus one, and c is minus six.","live":["l9"],"does":[[424.8926041666667,"heading_d is shown on the screen, written out."],[426.35510416666665,"axes3 is shown on the screen, written out."],[427.79510416666665,"curve5 is shown on the screen, written out."],[431.95110416666665,"sub0 is shown on the screen, written out."]]},{"start":436.2146041666667,"say":"Substitute. Minus b is plus one. b squared is one, and four a c is minus twenty four, so under the root we have one plus twenty four.","live":["l9","sub0","axes3","heading_d","curve5"],"does":[[436.5341041666667,"sub1 is shown on the screen, written out."]]},{"start":447.3276041666667,"say":"One plus twenty four is twenty five. And the square root of twenty five is exactly five, so the awkward part disappears.","live":["l9","sub0","sub1","axes3","heading_d","curve5"],"does":[[448.91810416666664,"sub1 becomes \"$x = frac(1 plus.minus sqrt(25), 2)$\"."],[451.8901041666667,"sub1 becomes \"$x = frac(1 plus.minus 5, 2)$\"."]]},{"start":455.5431041666667,"say":"So x is one plus five over two, or one minus five over two. Three, or minus two. The formula lands on the same two crossings the picture showed us at the start.","live":null,"does":[[455.89110416666665,"sub2 is shown on the screen, written out."],[464.64510416666667,"c_left is shown on the screen, written out."],[464.82560363076095,"c_right is shown on the screen, written out."]]},{"start":467.53260416666666,"say":"It worked here, and it will work where nothing factors, because we never once assumed the numbers were friendly. What is left is to read the formula properly, and everything interesting is hiding under that square root.","live":["l9","sub0","sub1","sub2","axes3","heading_d","curve5","c_left","c_right"],"does":[[477.3891041666666,"l9 (the \"sqrt(b^2 - 4 a c)\" part) is indicated — a transient flash."],[479.4166041666667,"axes3 is hidden from the screen — left the board."],[479.4166041666667,"curve5 is hidden from the screen — axes3 left the board."],[479.4166041666667,"c_left is hidden from the screen — axes3 left the board."],[479.4166041666667,"c_right is hidden from the screen — axes3 left the board."],[479.4166041666667,"heading_d is hidden from the screen — left the board."],[479.4166041666667,"l9 is hidden from the screen — left the board."],[479.4166041666667,"sub0 is hidden from the screen — left the board."],[479.4166041666667,"sub1 is hidden from the screen — left the board."],[479.4166041666667,"sub2 is hidden from the screen — left the board."]]}]},{"title":"What the Discriminant Decides","start":480.45827083333336,"end":688.9385,"objects":{"axes":"an Axes (x_range=(-1.0, 3.0), y_range=(-2.5, 3.5), x_ticks_every=1.0)","axes4":"an Axes (x_range=(-1.6, 3.6), y_range=(-2.5, 6.0), x_ticks_every=1.0)","axes_2":"an Axes (x_range=(-1.0, 3.0), y_range=(-2.5, 3.5), x_ticks_every=1.0)","axes_3":"an Axes (x_range=(-1.0, 3.0), y_range=(-2.5, 3.5), x_ticks_every=1.0)","cap_neg":"a Math [text] that says \"$b^2 - 4 a c < 0$\"","cap_pos":"a Math [text] that says \"$b^2 - 4 a c > 0$\"","cap_zero":"a Math [text] that says \"$b^2 - 4 a c = 0$\"","curve4":"a FunctionPlot [blue] drawn in axes4 (function=<function>, x_range=(-1.2, 3.2), y_offset=<VariableNumber shift = 1.0>)","d1":"a Math [text] that says \"$x^2 - 2 x + 2 = 0$\"","d2":"a Math [text] that says \"$x = frac(2 plus.minus sqrt(4 - 8), 2)$\"","d3":"a Math [text] that says \"$x = 1 plus.minus i$\"","disc4":"a Math [text] that says \"$b^2 - 4 a c = 4 - 4 c$\"","dnum":"a VariableNumber (initial_value=4.0, format_spec='.1f')","formula4":"a Math [text] that says \"$x = frac(- b, 2 a) plus.minus frac(sqrt(b^2 - 4 a c), 2 a)$\"","heading1":"a Heading that says \"What the Discriminant Decides\"","heading2":"a Heading that says \"Three Cases\"","heading3":"a Heading that says \"When Nothing Is Real\"","heading4":"a Heading that says \"What to Carry Away\"","neg_curve":"a FunctionPlot [blue] drawn in axes_3 (function=<function>, x_range=(-0.55, 2.55))","neg_sym":"a Line [gray] drawn in axes_3 (start=(1.0, -2.2), end=(1.0, 3.2), dashed=True)","pos_curve":"a FunctionPlot [blue] drawn in axes (function=<function>, x_range=(-0.55, 2.55))","pos_left":"a PlotPoint [red] drawn in axes (target='pos_curve', x=-0.41421356237309515)","pos_right":"a PlotPoint [red] drawn in axes (target='pos_curve', x=2.414213562373095)","quad4":"a Math [text] that says \"$x^2 - 2 x + c = 0$\"","r_left":"a PlotPoint [red] drawn in axes4 (target='curve4', x=(1.0 - sqrt((1.0 - shift))))","r_right":"a PlotPoint [red] drawn in axes4 (target='curve4', x=(1.0 + sqrt((1.0 - shift))))","readout":"a Point [text] labelled \"b^2 - 4 a c = 4.0\" drawn in axes4 (location=(2.7, 5.0), show_marker=False)","recap":"a Block [text] that says \"The formula is completing the square, done once with letters. $frac(- b, 2 a)$ is the centre: the axis of symmetry. $b^2 - 4 a c$ decides: positive gives two roots, zero gives one, negative gives a conjugate pair.\"","shift":"a VariableNumber","vsym4":"a Line [gray] drawn in axes4 (start=(1.0, -2.2), end=(1.0, 5.6), dashed=True)","zero_curve":"a FunctionPlot [blue] drawn in axes_2 (function=<function>, x_range=(-0.55, 2.55))","zero_root":"a PlotPoint [red] drawn in axes_2 (target='zero_curve', x=1.0)"},"beats":[{"start":480.45827083333336,"say":"Here is one parabola that I can slide up and down. Its equation is x squared minus two x plus c, and c is the only number I am going to touch.","live":[],"does":[[480.45827083333336,"heading1 is shown on the screen, written out."],[480.45827083333336,"axes4 is shown on the screen, written out."],[481.14327083333336,"curve4 is shown on the screen, written out."],[484.2432708333334,"axes4 moves to a new place on the board."],[484.2432708333334,"quad4 is shown on the screen, written out."]]},{"start":491.37977083333334,"say":"At c equals zero it crosses the axis in two places, at zero and at two. The dashed line is the axis of symmetry, halfway between them, at x equals one.","live":["quad4","axes4","heading1","curve4"],"does":[[494.9552708333334,"r_left is shown on the screen, written out."],[495.2960973546776,"r_right is shown on the screen, written out."],[498.26427083333334,"vsym4 is shown on the screen, written out."]]},{"start":504.39077083333336,"say":"For this family a is one and b is minus two, so b squared minus four a c is four minus four c. At c equals zero that reads four, a positive number, and I will keep it on the screen while c moves.","live":["quad4","axes4","heading1","curve4","r_left","r_right","vsym4"],"does":[[505.20327083333336,"disc4 is shown on the screen, written out."],[514.7352708333334,"readout is shown on the screen, written out."]]},{"start":520.3277708333334,"say":"Now lift the parabola. As c grows, that number shrinks, and the two crossings walk in toward the dashed line together.","live":["quad4","disc4","axes4","heading1","curve4","r_left","r_right","vsym4","readout"],"does":[[521.0362708333333,"curve4 is redrawn as the numbers it depends on change."],[521.0362708333333,"r_left is redrawn as the numbers it depends on change."],[521.0362708333333,"r_right is redrawn as the numbers it depends on change."],[521.0362708333333,"readout is redrawn as the numbers it depends on change."],[521.0362708333333,"shift ticks to 0.75."],[521.0362708333333,"dnum ticks to 1.0."]]},{"start":529.6697708333334,"say":"Keep going. At c equals one it is exactly zero, and the two crossings arrive at the same place. The parabola touches the axis there and does not pass through it.","live":null,"does":[[530.0182708333333,"curve4 is redrawn as the numbers it depends on change."],[530.0182708333333,"r_left is redrawn as the numbers it depends on change."],[530.0182708333333,"r_right is redrawn as the numbers it depends on change."],[530.0182708333333,"readout is redrawn as the numbers it depends on change."],[530.0182708333333,"shift ticks to 1.0."],[530.0182708333333,"dnum ticks to 0.0."],[535.6602708333334,"r_left is indicated — a transient flash."],[535.8445656778157,"r_right is indicated — a transient flash."]]},{"start":541.8797708333334,"say":"One root, or a repeated root, depending on how you like to count. And if I lifted it any further there would be nothing left on the axis to mark, because four minus four c would be negative. Let me put all three possibilities beside each other.","live":null,"does":[[556.8327708333334,"axes4 is hidden from the screen — left the board."],[556.8327708333334,"curve4 is hidden from the screen — axes4 left the board."],[556.8327708333334,"r_left is hidden from the screen — axes4 left the board."],[556.8327708333334,"r_right is hidden from the screen — axes4 left the board."],[556.8327708333334,"vsym4 is hidden from the screen — axes4 left the board."],[556.8327708333334,"readout is hidden from the screen — axes4 left the board."],[556.8327708333334,"disc4 is hidden from the screen — left the board."],[556.8327708333334,"heading1 is hidden from the screen — left the board."],[556.8327708333334,"quad4 is hidden from the screen — left the board."]]},{"start":558.0327708333334,"say":"Three parabolas of the same shape, three values of c. On the left the discriminant is positive. Two crossings, and two real roots.","live":[],"does":[[558.0327708333334,"heading2 is shown on the screen, written out."],[562.8162708333334,"axes is shown on the screen, written out."],[562.8162708333334,"pos_curve is shown on the screen, written out."],[563.9192708333334,"cap_pos is shown on the screen, written out."],[565.3242708333333,"pos_left is shown on the screen, written out."],[565.5062806548689,"pos_right is shown on the screen, written out."]]},{"start":568.1177708333333,"say":"In the middle it is zero. The turning point of the parabola sits exactly on the axis, and the two roots have collapsed into one.","live":["axes","cap_pos","heading2","pos_curve","pos_left","pos_right"],"does":[[568.7452708333334,"axes_2 is shown on the screen, written out."],[568.7452708333334,"zero_curve is shown on the screen, written out."],[569.3372708333334,"cap_zero is shown on the screen, written out."],[572.0302708333334,"zero_root is shown on the screen, written out."]]},{"start":577.0772708333334,"say":"On the right it is negative. The parabola clears the axis completely, so there is no real number at all that you could put in for x and get zero out.","live":["axes","cap_pos","axes_2","cap_zero","heading2","pos_curve","pos_left","pos_right","zero_curve","zero_root"],"does":[[577.7732708333333,"axes_3 is shown on the screen, written out."],[577.7732708333333,"neg_curve is shown on the screen, written out."],[578.3542708333333,"cap_neg is shown on the screen, written out."],[585.6567708333333,"axes_3 moves to a new place on the board."],[585.6567708333333,"axes is hidden from the screen — left the board."],[585.6567708333333,"pos_curve is hidden from the screen — axes left the board."],[585.6567708333333,"pos_left is hidden from the screen — axes left the board."],[585.6567708333333,"pos_right is hidden from the screen — axes left the board."],[585.6567708333333,"axes_2 is hidden from the screen — left the board."],[585.6567708333333,"zero_curve is hidden from the screen — axes_2 left the board."],[585.6567708333333,"zero_root is hidden from the screen — axes_2 left the board."],[585.6567708333333,"cap_neg is hidden from the screen — left the board."],[585.6567708333333,"cap_pos is hidden from the screen — left the board."],[585.6567708333333,"cap_zero is hidden from the screen — left the board."],[585.6567708333333,"heading2 is hidden from the screen — left the board."]]},{"start":586.8567708333334,"say":"But no real root is not the same as no root. Take that third case on its own. x squared minus two x plus two equals zero, so a is one, b is minus two, and c is two.","live":["axes_3","neg_curve"],"does":[[586.8567708333334,"heading3 is shown on the screen, written out."],[590.2582708333333,"d1 is shown on the screen, written out."],[599.2332708333333,"d2 is shown on the screen, written out."]]},{"start":601.9577708333334,"say":"Under the root we get four minus eight, which is minus four. No real number squares to something negative, so this is where we widen what counts as a number: the square root of minus four is two i, where i squared is minus one.","live":["axes_3","neg_curve","d1","d2","heading3"],"does":[[602.3062708333333,"d2 becomes \"$x = frac(2 plus.minus sqrt(-4), 2)$\"."],[609.8762708333334,"d2 becomes \"$x = frac(2 plus.minus 2 i, 2)$\"."]]},{"start":617.6152708333334,"say":"Divide the two through, and the roots are one plus i and one minus i. A conjugate pair, and look where they sit: either side of x equals one, which is the axis of symmetry, exactly as the real pairs did.","live":null,"does":[[617.9632708333334,"d3 is shown on the screen, written out."],[626.1252708333334,"neg_sym is shown on the screen, written out."]]},{"start":632.7397708333333,"say":"So the same formula answers all three cases, and it always answers them in the same shape: a centre, and a distance either side of it.","live":["axes_3","neg_curve","d1","d2","d3","heading3","neg_sym"],"does":[[641.5052708333334,"axes_3 is hidden from the screen — left the board."],[641.5052708333334,"neg_curve is hidden from the screen — axes_3 left the board."],[641.5052708333334,"neg_sym is hidden from the screen — axes_3 left the board."],[641.5052708333334,"d1 is hidden from the screen — left the board."],[641.5052708333334,"d2 is hidden from the screen — left the board."],[641.5052708333334,"d3 is hidden from the screen — left the board."],[641.5052708333334,"heading3 is hidden from the screen — left the board."]]},{"start":642.1052708333334,"say":"Split the formula that way and it reads out loud. Minus b over two a is the centre, the axis of symmetry. The square root of the discriminant over two a is how far out the roots lie.","live":[],"does":[[642.1052708333334,"heading4 is shown on the screen, written out."],[642.5812708333333,"formula4 is shown on the screen, written out."],[647.7822708333333,"formula4 (the \"frac(- b, 2 a)\" part) is emphasized."],[654.1912708333334,"formula4 (the \"frac(- b, 2 a)\" part) is no longer emphasized."],[654.1912708333334,"formula4 (the \"frac(sqrt(b^2 - 4 a c), 2 a)\" part) is emphasized."],[656.1302708333334,"formula4 (the \"frac(sqrt(b^2 - 4 a c), 2 a)\" part) is no longer emphasized."]]},{"start":656.7302708333334,"say":"Three things to carry away. The formula is nothing more than completing the square, done once with letters instead of numbers.","live":["formula4","heading4"],"does":[[657.2062708333333,"recap is shown on the screen, written out."],[660.4342708333334,"recap (the \"completing the square\" part) is emphasized."]]},{"start":665.7247708333333,"say":"Minus b over two a is the centre of the picture, and the two roots are always symmetric about it.","live":["formula4","recap","heading4"],"does":[[668.1972708333333,"recap (the \"completing the square\" part) is no longer emphasized."],[668.1972708333333,"recap (the \"the axis of symmetry\" part) is emphasized."]]},{"start":672.7327708333333,"say":"And b squared minus four a c decides the whole story. Positive, two real roots. Zero, one. Negative, a conjugate pair, and a parabola that never touches the axis at all.","live":null,"does":[[675.3682708333333,"recap (the \"decides\" part) is emphasized."],[675.3682708333333,"recap (the \"the axis of symmetry\" part) is no longer emphasized."],[682.4732708333333,"recap (the \"decides\" part) is no longer emphasized."],[683.4252708333333,"A box is drawn around formula4."],[687.8968333333333,"formula4 is hidden from the screen — left the board."],[687.8968333333333,"heading4 is hidden from the screen — left the board."],[687.8968333333333,"recap is hidden from the screen — left the board."]]}]}]},"durationSeconds":689,"chapters":[{"title":"What a Root Is","startSeconds":0,"narration":"Here is a question that covers every quadratic equation there is. Solve a x squared plus b x plus c equals zero, for x, whatever the three numbers a, b and c happen to be. By the end we will have one line of algebra that answers it every single time. A quadratic is easiest to meet as a picture. This blue curve is y equals x squared minus x minus six. One bend, opening upward, and that shape is called a parabola. Solving the equation means finding the x values where y is zero, and a height of zero means the curve is sitting on the horizontal axis. This one meets the axis in two places, here and here. Those two places are at minus two and at three. Those numbers are the roots of the equation, and they are what the whole lecture is about. For this particular one we can find them with no machinery at all. The left-hand side factors: x squared minus x minus six is x minus three, times x plus two. And a product can only be zero when one of the two factors is zero. So x equals three, or x equals minus two, which is exactly what the picture already told us. Now change the three numbers and try again. This green curve is y equals x squared minus four x plus one, and it crosses the axis twice as well. But try factoring this one. Nothing with whole numbers works, and the reason is visible on the picture: the crossings are not whole numbers. They are two minus the square root of three, and two plus the square root of three. Guessing factors is a trick that works when it works. What we want is something that never fails, and here it is: the quadratic formula. Read it once. Minus b at the front. Then plus or minus a square root. Under that root, b squared minus four a c. And the whole thing divided by two a. That piece under the root has a name of its own: the discriminant. Whether a parabola crosses the axis twice, touches it once, or misses it entirely is decided by that one expression, and the last part of the lecture is about exactly that."},{"title":"Completing the Square","startSeconds":142.6826875,"narration":"Everything rests on one small fact: a square is easy to undo. If something squared is sixteen, then that something is four, or minus four. So the plan is to force any quadratic into exactly that shape. Here is a concrete one to practise on. x squared plus six x minus seven equals zero. It does happen to factor, but ignore that and watch a different route. Move the seven out of the way first, so that the two x terms stand alone. x squared plus six x equals seven. Now draw x squared as an actual square, with side x. And draw six x as two rectangles, three wide and x tall, laid against two of its edges. Look at the shape those three pieces make. It is a big square with one corner missing, and the missing corner is three by three. Its area is nine. So add nine to the left-hand side. And add nine to the right as well, to keep the equation true. That gives x squared plus six x plus nine equals sixteen. And the left-hand side is now a completed square. The picture says the same thing: the whole edge along the bottom is x plus three, so the area is x plus three, all squared, and that area is sixteen. Before finishing the algebra, put the parabola on the screen. y equals x squared plus six x minus seven. Its two crossings are the answers we are about to compute. Take the square root of both sides. Sixteen has two square roots, four and minus four, so plus or minus arrives right here, and this is the only place in the whole method that it comes from. Then subtract three from both sides. Minus three plus four is one, and minus three minus four is minus seven. There they are on the curve. And notice where those two roots sit. The parabola turns at x equals minus three, and the roots are four units to the left of that and four units to the right. The plus or minus is the symmetry of the curve, written as one symbol. That is the entire trick, and nothing in it cared that the numbers were six and seven. So let us run it again with letters."},{"title":"The Same Steps, With Letters","startSeconds":289.66210416666667,"narration":"Same five moves, with a, b and c standing in for whatever the numbers are. Start from the general equation, with a not equal to zero, because otherwise there is no x squared term and no quadratic. The trick needs the x squared term to have nothing in front of it, so divide every single term by a. That is allowed, precisely because a is not zero. Move the constant across to the right, exactly as we moved the seven before. Now the key step. Last time we added three squared, and three was half of six, the coefficient of x. Here the coefficient of x is b over a, half of that is b over two a, and so what we add to both sides is b over two a, all squared. The left-hand side is now a completed square, and the number inside the bracket is that same half: x plus b over two a, all squared. On the right, b over two a squared is b squared over four a squared. Put the right-hand side over a single denominator. Multiply c over a top and bottom by four a, and the two fractions combine into one. Look at that numerator. b squared minus four a c. Nobody put it there; it fell out of the arithmetic, and it is the expression we will read the geometry off in the last part. Take the square root of both sides. The square root of four a squared is two a, and a square root comes with two signs, so plus or minus appears here, once, and never again. Then subtract b over two a from both sides, and x stands alone. Both of those terms already have the same denominator, two a, so write them as a single fraction. And that is the quadratic formula. There it is on its own. Minus b, plus or minus the square root of b squared minus four a c, all divided by two a. Nothing in that line was guessed. One warning before we test it. The letters have to come from an equation with zero on the right. If yours is not in that shape yet, put it in that shape first, and keep every minus sign as you read the numbers off. Test it on the very first parabola. y equals x squared minus x minus six, so a is one, b is minus one, and c is minus six. Substitute. Minus b is plus one. b squared is one, and four a c is minus twenty four, so under the root we have one plus twenty four. One plus twenty four is twenty five. And the square root of twenty five is exactly five, so the awkward part disappears. So x is one plus five over two, or one minus five over two. Three, or minus two. The formula lands on the same two crossings the picture showed us at the start. It worked here, and it will work where nothing factors, because we never once assumed the numbers were friendly. What is left is to read the formula properly, and everything interesting is hiding under that square root."},{"title":"What the Discriminant Decides","startSeconds":480.45827083333336,"narration":"Here is one parabola that I can slide up and down. Its equation is x squared minus two x plus c, and c is the only number I am going to touch. At c equals zero it crosses the axis in two places, at zero and at two. The dashed line is the axis of symmetry, halfway between them, at x equals one. For this family a is one and b is minus two, so b squared minus four a c is four minus four c. At c equals zero that reads four, a positive number, and I will keep it on the screen while c moves. Now lift the parabola. As c grows, that number shrinks, and the two crossings walk in toward the dashed line together. Keep going. At c equals one it is exactly zero, and the two crossings arrive at the same place. The parabola touches the axis there and does not pass through it. One root, or a repeated root, depending on how you like to count. And if I lifted it any further there would be nothing left on the axis to mark, because four minus four c would be negative. Let me put all three possibilities beside each other. Three parabolas of the same shape, three values of c. On the left the discriminant is positive. Two crossings, and two real roots. In the middle it is zero. The turning point of the parabola sits exactly on the axis, and the two roots have collapsed into one. On the right it is negative. The parabola clears the axis completely, so there is no real number at all that you could put in for x and get zero out. But no real root is not the same as no root. Take that third case on its own. x squared minus two x plus two equals zero, so a is one, b is minus two, and c is two. Under the root we get four minus eight, which is minus four. No real number squares to something negative, so this is where we widen what counts as a number: the square root of minus four is two i, where i squared is minus one. Divide the two through, and the roots are one plus i and one minus i. A conjugate pair, and look where they sit: either side of x equals one, which is the axis of symmetry, exactly as the real pairs did. So the same formula answers all three cases, and it always answers them in the same shape: a centre, and a distance either side of it. Split the formula that way and it reads out loud. Minus b over two a is the centre, the axis of symmetry. The square root of the discriminant over two a is how far out the roots lie. Three things to carry away. The formula is nothing more than completing the square, done once with letters instead of numbers. Minus b over two a is the centre of the picture, and the two roots are always symmetric about it. And b squared minus four a c decides the whole story. Positive, two real roots. Zero, one. Negative, a conjugate pair, and a parabola that never touches the axis at all."}]}}