Lesson 13

Completing the Square (Part 2)

Lesson Narrative

In this lesson, students learn that completing the square can be used to solve any quadratic equation, including equations that involve rational numbers that are not integers. Students notice that the process of completing the square is the same when the equations involve messier numbers as when they have simple integers, but the calculations may be more time consuming and prone to error. An error-analysis activity highlights some common errors related to completing the square.

Although any equation can be solved by completing the square, equations that are really difficult to solve by this method are not included here. Students will solve such equations when they have access to the quadratic formula. What is important in this lesson is to recognize that putting a quadratic equation in the form of $$(x+p)^2 =q$$ allows them to solve it, but there are cases in which doing so may not always be the most efficient strategy.

Completing the square for quadratic expressions that are more elaborate encourages students to look for and make use of the same structure that helped them when they were working with less complicated expressions (MP7).

Learning Goals

Teacher Facing

• Express any quadratic equation in the form $(x+p)^2=q$ and solve the equation by finding square roots.
• Generalize (orally) a process for completing the square to express any quadratic equation in the form $(x+p)^2=q$.
• Solve quadratic equations in which the squared term has a coefficient of 1 by completing the square.

Student Facing

• Let’s solve some harder quadratic equations.

Student Facing

• When given a quadratic equation in which the coefficient of the squared term is 1, I can solve it by completing the square.

Glossary Entries

• completing the square

Completing the square in a quadratic expression means transforming it into the form $$a(x+p)^2-q$$, where $$a$$, $$p$$, and $$q$$ are constants.

Completing the square in a quadratic equation means transforming into the form $$a(x+p)^2=q$$.

• perfect square

A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.

• rational number

A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into $$b$$ equal parts and finding the point that is $$a$$ of them from 0. We can always write a fraction in the form $$\frac{a}{b}$$ where $$a$$ and $$b$$ are whole numbers, with $$b$$ not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as $$\frac{7}{10}$$.

The numbers $$3$$, $$\text-\frac34$$, and $$6.7$$ are all rational numbers. The numbers $$\pi$$ and $$\text-\sqrt{2}$$ are not rational numbers, because they cannot be written as fractions or their opposites.