Quadratic Equations and Functions
Solve Quadratic Equations by Completing the Square
Learning Objectives
By the end of this section, you will be able to:
- Complete the square of a binomial expression
- Solve quadratic equations of the form by completing the square
- Solve quadratic equations of the form by completing the square
Before you get started, take this readiness quiz.
So far we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called completing the square, which is important for our work on conics later.
Complete the Square of a Binomial Expression
In the last section, we were able to use the Square Root Property to solve the equation (y − 7)2 = 12 because the left side was a perfect square.
We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form in order to use the Square Root Property.
What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?
Let’s look at two examples to help us recognize the patterns.
We restate the patterns here for reference.
If a and b are real numbers,
We can use this pattern to “make” a perfect square.
We will start with the expression x2 + 6x. Since there is a plus sign between the two terms, we will use the (a + b)2 pattern, a2 + 2ab + b2 = (a + b)2.
We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find b. But first we start with determining a. Notice that the first term of x2 + 6x is a square, x2. This tells us that a = x.
What number, b, when multiplied with 2x gives 6x? It would have to be 3, which is So b = 3.
Now to complete the perfect square trinomial, we will find the last term by squaring b, which is 32 = 9.
We can now factor.
So we found that adding 9 to x2 + 6x ‘completes the square’, and we write it as (x + 3)2.
- Identify b, the coefficient of x.
- Find the number to complete the square.
- Add the to x2 + bx.
- Factor the perfect square trinomial, writing it as a binomial squared.
Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
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| The coefficient of is −26. | |
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