The central idea in the derivation of the quadratic formula is a step called completing the square, which uses the factoring formula for the square of a sum. To complete the square, we search for a number such that when we add it to x and square the sum, it will absorb the linear term.

Derivation

We begin with our initial quadratic equation, subtract c from both sides, and then divide both sides by a.

At this stage, we are ready to put our equation in the proper form to complete the square, using this value in place of the empty square:

Below, we rewrite the left side to be in the form that we desire by multiplying and dividing the coefficient of the linear term by 2. To finish completing the square, we add the square of the quantity in the parentheses to both sides. This makes the right side the square of a sum.

In the third line, we have a squared quantity equal to the left side that we have simplified with a common denominator. Then we take the square root of both sides; we have a two roots, which are shown by the ± sign. With a few more simplifications, we finish the rest to give us our quadratic formula.