In earlier lessons, we worked with perfect squares such as and . We learned that their equivalent expressions in standard form follow a predictable pattern:

• In general, can be written as .
• If a quadratic expression of the form is a perfect square, and the value of is 1, then the value of is , and the value of is for some value of .

In this lesson, the variables in the factors being squared had coefficients other than 1, for example and . Their equivalent expressions in standard form also followed the same pattern we saw earlier.

squared factor	standard form

In general, can be written as:

If a quadratic expression is of the form , then:

• The value of is .
• The value of is .
• The value of is .

We can use this pattern to help us complete the square and solve equations when the squared term has a coefficient other than 1—for example, .

What constant term can we add to make the expression on the left of the equal sign a perfect square? And how do we write this expression as a squared factor?

• , which is , so , and the squared factor could be .
• 40 is equal to , so , or . This means that .
• If is , then , or .
• So the expression is a perfect square and is equivalent to .

Let’s solve the equation by completing the square!

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