Lesson 7

Inequivalent Equations

• Let’s see what happens when we square each side of an equation.

Problem 1

Noah solved the equation $$5x^2=45$$. Here are his steps:

\begin{align} 5x^2 &= 45 \\ x^2 &= 9 \\ x &= 3 \\ \end{align}

Do you agree with Noah? Explain your reasoning.

Problem 2

Find the solution(s) to each equation, or explain why there is no solution.

1. $$\sqrt{x+4}+7=5$$
2. $$\sqrt{47-x}-2 = 4$$
3. $$\frac12 \sqrt{20+x} = 5$$

Problem 3

Which is a solution to the equation $$\sqrt{5-x}+13=4$$?

A:

86

B:

81

C:

9

D:

The equation has no solution.

Problem 4

Select all expressions that are equal to $$\frac{1}{(\sqrt2)^5}$$.

A:

$$\text- \frac{5}{\sqrt2}$$

B:

$$\frac{1}{\sqrt{2^5}}$$

C:

$$\frac{1}{\sqrt{32}}$$

D:

$$\text- (\sqrt2)^5$$

E:

$$\text- 2^{\frac52}$$

F:

$$2^{\text- \frac52}$$

(From Unit 3, Lesson 5.)

Problem 5

Which are the solutions to the equation $$x^2=36$$?

A:

6 only

B:

-6 only

C:

6 and -6

D:

This equation has no solutions.

(From Unit 3, Lesson 6.)

Problem 6

Here is a graph of $$y=x^2$$.

1. Use the graph to estimate all solutions to the equation $$x^2=3$$.
2. If you square your estimates, what number should they be close to?
3. Square your estimates. How close did you get to this number?
(From Unit 3, Lesson 6.)

Problem 7

The polynomial function $$q(x)=3x^3+11x^2-14x-40$$ has a known factor of $$(3x + 5)$$. Rewrite $$q(x)$$ as the product of linear factors.

(From Unit 2, Lesson 12.)