Completing the Square Methods of Solving Quadratic Equations You have learned three methods for solving equations of the form

When , you can solve by using the square root property For example: If the expression is factorable, then you can use the Product Property of Zero For example: You can use the Quadratic Formula Completing the Square You have one more method of solving to learn

First, you must learn a new technique called completing the square To begin to understand this technique, you will start by performing the following multiplications and looking for a pattern 1) 2) 3) 4) Completing the Square

The results are as follows 1) 2) 3) 4) Do you detect a pattern? Sometimes this pattern is difficult to detect unless we choose not to simplify the multiplication of the numbers involved

Here are the results again, but without having carried out multiplication Completing the Square Can you see the pattern? 1) 2) 3) 4)

If you have an expression , how would you describe the pattern (what happens to a when you square the binomial)? If the terms are x and a, then you might say that 1. You square the first term to get 2. Then you multiply the second term by 2 and then by x to get 3. Finally, you square the second term to get The result is

Completing the Square Now that you know that , you can tell at a glance whether a quadratic expression is a perfect square binomial For example, is a perfect square binomial because, using the pattern of , we have that so that , and is the last term Therefore, we can write (that is, we can factor as) Completing the Square

Example: is the expression a perfect square binomial? Using the pattern , we see that so that But Therefore, this is not a perfect square binomial; it cannot be factored as Guided Practice Tell whether the expression is a perfect square binomial. If it is, write it as a perfect square binomial.

a) b) c) d) Guided Practice Tell whether the expression is a perfect square binomial. If it is, write it as a perfect square binomial. a)

b) not a perfect square binomial c) d) not a perfect square binomial Completing the Square The technique of completing the square requires us to change a quadratic expression that is not a perfect square binomial, into an expression that includes a perfect square binomial Using numbers, note that 17 is not a perfect square

But and since , then We have taken a number that is not a perfect square, and rewritten it as an expression that includes a perfect square Completing the Square To do this with a quadratic expression of the form that is not a perfect, we must first determine using the value of b what would have to be added to make it a perfect square For example, you saw in the guided practice that is not a perfect

square because if then , but So is a perfect square, but is not a perfect square Completing the Square However, we can use the inverse property for addition to fix this expression so that it contains a perfect square We do this by adding zero, but in the form of Since , then we can rewrite the expression as This is a true statement since we have added zero

Now, We have written the expression so that it includes a perfect square; this is completing the square Completing the Square Lets develop a consistent method for completing the square If , how are a, b, and c related? We saw that and that Since we will use b to determine c, then we note that and therefore

This would mean that Completing the Square Examples Now lets use to work some examples Example: complete the square for Here, so Therefore, is a perfect square Rewrite the expression by adding and subtracting 4 Complete the square and combine like terms to get

Completing the Square Examples Now lets use to work some examples Example: complete the square for Here, so Therefore, is a perfect square Rewrite the expression by adding and subtracting 64 Complete the square and combine like terms to get

Completing the Square Examples Up to now, the examples have been chosen so that b is an even number, but what is b is odd? Example: complete the square for Here we have so Therefore, is a perfect square Rewrite the expression, adding and subtracting Complete the square and combine like terms to get

Completing the Square Examples Example: complete the square for Here we have so Therefore, is a perfect square Rewrite the expression, adding and subtracting Complete the square and combine like terms to get Guided Practice Complete the square.

a) b) c) d) Guided Practice Complete the square. a) b)

c) d) Exercise 4.7a Handout