Completing the Square

Examples, solutions, and videos to help Algebra I students learn how to rewrite quadratic expressions given in standard form, ax² + bx + c, in the equivalent completed-square form, a(x-h)² + k, and recognize cases for which factored or completed-square form is most efficient to use.

New York State Common Core Math Algebra I, Module 4, Lesson 11, Lesson 12

Worksheets for Algebra 1

Lesson 11 Summary

Just as factoring a quadratic expression can be useful for solving a quadratic equation, completing the square also provides a form that facilitates solving a quadratic equation.

Lesson 11 Opening Exercise:

In lesson 11, we will look at completing the square for expressions where the leading coefficient a = 1.

Rewrite the following perfect square quadratic expressions in standard form.
(x + 10)²
(x + 6)²

Example 1
Rewrite the following standard form quadratic expressions as perfect squares.
x² - 8x + 16
x² + 12x + 36

Example 2
Find an expression equivalent to x² + 10x + 36 that includes a perfect square binomial.

Exercises 1 - 10

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Lesson 12

In lesson 12, we will look at completing the square for expressions where the leading coefficient a ≠ 1.

Example 1:
Complete the square for
2x² + 16x + 3

Example 2:
A certain business is marketing their product and has collected data on sales and prices for the past few years. They determined that when they raised the selling price of the product, the number of sales went down. The cost of producing a single item is $10. a. Using the data they collected in this table, determine a linear expression to represent the quantity sold, q.
b. Now find an expression to represent the profit function, P.

Exercises 1 - 6: Rewrite each expression by completing the square.
3x² + 12x - 8
4p² - 12p +13

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