# Lesson 4

Solving Quadratic Equations with the Zero Product Property

### Problem 1

If the equation $$(x+10) x=0$$ is true, which statement is also true according to the zero product property?

A:

only $$x = 0$$

B:

either $$x = 0$$ or $$x + 10 = 0$$

C:

either $$x^2 = 0$$ or $$10x=0$$

D:

only $$x + 10 = 0$$

### Problem 2

What are the solutions to the equation $$(10-x)(3x-9)=0$$?

A:

-10 and 3

B:

-10 and 9

C:

10 and 3

D:

10 and 9

### Problem 3

Solve each equation.

1. $$(x-6)(x+5)=0$$
2. $$(x-3)(\frac23 x - 6)=0$$
3. $$(\text-3x-15)(x+7)=0$$

### Problem 4

Consider the expressions $$(x-4)(3x-6)$$ and $$3x^2 - 18x + 24$$.

Show that the two expressions define the same function.

### Problem 5

Kiran saw that if the equation $$(x+2)(x-4)=0$$ is true, then, by the zero product property, either $$x+2$$ is 0 or $$x-4$$ is 0. He then reasoned that, if $$(x+2)(x-4)=72$$ is true, then either $$x+2$$ is equal to 72 or $$x-4$$ is equal to 72.

Explain why Kiran’s conclusion is incorrect.

### Problem 6

Andre wants to solve the equation $$5x^2-4x-18=20$$. He uses a graphing calculator to graph $$y=5x^2-4x-18$$ and $$y=20$$ and finds that the graphs cross at the points $$(\text-2.39, 20)$$ and $$(3.19, 20)$$.

1. Substitute each $$x$$-value Andre found into the expression $$5x^2-4x-18$$. Then evaluate the expression.
2. Why did neither solution make $$5x^2-4x-18$$ equal exactly 20?

### Solution

(From Unit 7, Lesson 2.)

### Problem 7

Select all the solutions to the equation $$7x^2 = 343$$.

A:

49

B:

$$\text-\sqrt{7}$$

C:

7

D:

-7

E:

$$\sqrt{49}$$

F:

$$\sqrt{\text- 49}$$

G:

$$\text- \sqrt{49}$$

### Solution

(From Unit 7, Lesson 3.)

### Problem 8

Here are two graphs that correspond to two patients, A and B. Each graph shows the amount of insulin, in micrograms (mcg) in a patient's body $$h$$ hours after receiving an injection. The amount of insulin in each patient decreases exponentially.

​​​​​​

Select all statements that are true about the insulin level of the two patients.

A:

After the injection, the patients have the same amount of insulin in their bodies.

B:

An equation for the micrograms of insulin, $$a$$, in Patient A's body $$h$$ hours after the injection is $$a = 200 \boldcdot \left(\frac{3}{5}\right)^h$$.

C:

The insulin in Patient A is decaying at a faster rate than in Patient B.

D:

After 3 hours, Patient A has more insulin in their body than Patient B.

E:

At some time between 2 and 3 hours, the patients have the same insulin level.

### Solution

(From Unit 5, Lesson 6.)

### Problem 9

Han says this pattern of dots can be represented by a quadratic relationship because the dots are arranged in a rectangle in each step.

Do you agree? Explain your reasoning.