# Lesson 9

Solving Quadratic Equations by Using Factored Form

### Lesson Narrative

In this lesson, students apply what they learned about transforming expressions into factored form to make sense of quadratic equations and persevere in solving them (MP1). They see that rearranging equations so that one side of the equal sign is 0, rewriting the expression in factored form, and then using the zero product property make it possible to solve equations that they previously could only solve by graphing. These steps also allow them to easily see—without graphing and without necessarily completing the solving process—the number of solutions that the equations have.

### Learning Goals

Teacher Facing

• Recognize that the number of solutions to a quadratic equation can be revealed when the equation is written as $\text {expression in factored form} = 0$.
• Use factored form and the zero product property to solve quadratic equations.

### Student Facing

• Let’s solve some quadratic equations that before now we could only solve by graphing.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can rearrange a quadratic equation to be written as $\text {expression in factored form}=0$ and find the solutions.
• I can recognize quadratic equations that have 0, 1, or 2 solutions when they are written in factored form.

Building On

Building Towards

### Glossary Entries

• coefficient

In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.

The coefficient of $$x$$ in the expression $$3x + 2$$ is $$3$$. The coefficient of $$p$$ in the expression $$5 + p$$ is 1.

• constant term

In an expression like $$5x + 2$$ the number 2 is called the constant term because it doesn't change when $$x$$ changes.

In the expression $$5x-8$$ the constant term is -8, because we think of the expression as $$5x + (\text-8)$$. In the expression $$12x-4$$ the constant term is -4.

• linear term

The linear term in a quadratic expression (In standard form) $$ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants, is the term $$bx$$. (If the expression is not in standard form, it may need to be rewritten in standard form first.)

• zero product property

The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.