# Lesson 5

How Many Solutions?

### Lesson Narrative

The work in this lesson builds on the idea that both graphing and rewriting quadratic equations in the form of $$\text{expression} =0$$ are useful strategies for solving equations. It also reinforces the ties between the zeros of a function and the horizontal intercepts of its graph, which students began exploring in an earlier unit.

Previously, to solve an equation such as $$\text-16t^2 + 8t + 10 = 32$$ by graphing, students would graph $$y=\text-16t^2 + 8t + 10$$ and $$y=32$$ and inspect where the parabolic graph and the horizontal line intersect.

Here, students learn another way to use graphs to solve equations and to anticipate the number of solutions. Instead of graphing two separate equations—one quadratic and one linear, students learn that they can solve by rearranging the equation into the form $$\text{expression} = 0$$, graphing the equation $$y= \text{expression}$$, and finding the horizontal intercepts. Why does this make sense?

• The $$x$$-coordinate of those intercepts produces a $$y$$-coordinate of 0, so they are the solutions to the equation $$\text{expression} = 0$$.
• The number of horizontal intercepts tells us the instances when the $$y$$-coordinate is 0, which tells us the number of solutions to the equation.

Later in the lesson, students think about why a quadratic equation that has an expression in factored form on one side of the equal sign but does not have 0 on the other side cannot be solved the same way as when the equation is $$\text{expression} = 0$$. They also notice that dividing each side of a quadratic equation by a variable is not reliable because it eliminates one of the solutions. As they explain why certain maneuvers are acceptable and others are not, students practice constructing logical arguments (MP3).

### Learning Goals

Teacher Facing

• Coordinate (orally) graphs with no horizontal intercepts, quadratic functions with no (real) zeros, and quadratic equations with no (real) solutions.
• Describe (orally and in writing) the relationship between the solutions to quadratic equations of the form $\text {expression}=0$ and the horizontal intercepts of the graph of the related function.
• Explain (orally and in writing) why dividing each side of a quadratic equation by a variable is not a reliable way to solve the equation.

### Student Facing

• Let’s use graphs to investigate quadratic equations that have two solutions, one solution, or no solutions.

### 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 explain why dividing by a variable to solve a quadratic equation is not a good strategy.
• I know that quadratic equations can have no solutions and can explain why there are none.

Building On