Select all expressions that are perfect squares. Explain how you know.
12.2
Activity
Complete the table so that each row has equivalent expressions that are perfect squares.
standard form
factored form
12.3
Activity
One technique for solving quadratic equations is called completing the square. Here are two examples of how Diego and Mai completed the square to solve the same equation.
Diego:
Mai:
Study the examples, then solve these equations by completing the square:
Student Lesson Summary
Turning an expression into a perfect square can be a good way to solve a quadratic equation. Suppose we wanted to solve .
The expression on the left, , is not a perfect square, but is a perfect square. Let’s transform that side of the equation into a perfect square (while keeping the equality of the two sides).
One helpful way to start is by first moving the constant that is not a perfect square out of the way. Let’s subtract 10 from each side:
And then add 49 to each side:
The left side is now a perfect square because it’s equivalent to or . Let’s rewrite it:
If a number squared is 9, the number has to be 3 or -3. Solve to finish up:
This method of solving quadratic equations is called completing the square. In general, perfect squares in standard form look like , so to complete the square, take half of the coefficient of the linear term and square it.
In the example, half of -14 is -7, and is 49. We wanted to make the left side To keep the equation true and maintain equality of the two sides of the equation, we added 49 to each side.
Completing the square is a method of rewriting a quadratic expression or equation.
A quadratic expression is rewritten in the form , where , , and are constants and
