Lesson 4

Solving Quadratic Equations with the Zero Product Property

  • Let’s find solutions to equations that contain products that equal zero.

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?
(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:

\(\text- \sqrt{49}\)

(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' body \(h\) hours after receiving an injection. The amount of insulin in each patient decreases exponentially.

Patient A

Graph of an exponential function, origin O, with grid. time (hours) and insulin (mcg).

​​​​​​

Patient B

Graph of a function.

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. 

(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.

Pattern of dots arranged in rectangles. Step 1 has 4 dots. Step 2 has 8 dots. Step 3 has 12 dots. Step 4 has 16 dots.
(From Unit 6, Lesson 2.)