Understanding the Quadratic Formula and How It Works

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.