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:
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:
1. Write the given equation in the standard form of the quadratic equation:
2. Factor out and divide both sides by the coefficient of x2 if it does not already equal 1:
Note: Dividing both sides by a cancels out a on the left side of the equation. On the right, .
3. Move the constant to the other side of the equation.
4. Add to both sides of the equation (to keep them equivalent):
Note: You do not have to simplify on the side with the perfect square trinomial, but it is necessary to on the side with the constant.
5. Factor the side with :
6. Solve for x:
Note: Always remember that the square root of a positive number has two answers, one positive and one negative!