Completing the square

Completing the square is a method used to solve quadratic equations. It can also be used to convert the general form of a quadratic, ax² + bx + c to the vertex form a(x - h)² + k

Generally, the goal behind completing the square is to create a perfect square trinomial from a quadratic. A perfect square trinomial is a trinomial that will factor into the square of a binomial. The square of a binomial is a binomial multiplied by itself.

Binomials of the form x + n, where n is some constant, are some of the easier binomials to work with. The square of x + n is x² + 2nx + n². As you can see, the coefficient of x is 2n.

For a quadratic of the form x² + bx + c, the coefficent of x is already b, so we only need to figure out the value of the constant c. If 2n = b, then n = . Therefore, the binomial used to complete the square is:

If we square this binomial, the perfect square trinomial needed to complete the square will equal:

or

Notice that there are cases where you will subtract . This is because if b is negative, then the constant in the binomial will need to be negative as well. is positive because any number squared is positive.

If we go back to the standard form of the quadratic equation, ax² + bx + c = 0, you will notice two differences between this perfect square trinomial and the quadratic:

1. The trinomial does not contain the coefficient a, or rather, a = 1

2. c is replaced by

The objective of completing the square is to satisfy the above two conditions. Let's look at each step in the process: