Why does the quadratic formula work the way it does?

Question

I use the quadratic formula all the time but I've never understood where it actually comes from. It looks so arbitrary with the plus-minus and the square root and the division by 2a. Can someone show me how you would derive it from scratch using completing the square?

Priya Sharma · Accepted Answer

The quadratic formula comes from completing the square on the general quadratic equation: a x 2 + b x + c = 0 , a ≠ 0. First divide everything by a: x 2 + b a x + c a = 0. Move the constant term: x 2 + b a x = − c a . To complete the square, take half of the coefficient of x and square it: ( b 2 a ) 2 . Add it to both sides: x 2 + b a x + ( b 2 a ) 2 = − c a + ( b 2 a ) 2 . The left side is now a square: ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 . Take the square root of both sides: x + b 2 a = ± b 2 − 4 a c 2 a . Finally subtract b 2 a: x = − b ± b 2 − 4 a c 2 a . So nothing in the formula is random. The ± comes from taking a square root, the 2 a comes from completing the square after dividing by a, and b 2 − 4 a c is the expression left under the square root.

Why does the quadratic formula work the way it does?

1 Answer

Why does the quadratic formula work the way it does?

1 Answer