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Previously, we saw that can be expanded to standard form as .
Write each expression in standard form:
Decide if each expression is a perfect square. If so, write an equivalent expression of the form . If not, suggest one change to turn it into a perfect square.
| standard form | squared factor |
|---|---|
Solve each equation by completing the square:
Here are three methods for solving .
Try to make sense of each method.
Method 1:
Method 2:
Method 3:
Once you understand the methods, use each method at least one time to solve these equations.
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 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:
or
If a quadratic expression is of the form , then:
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?
Let’s solve the equation by completing the square!
.