# 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.

- \(\sqrt{x+4}+7=5\)
- \(\sqrt{47-x}-2 = 4\)
- \(\frac12 \sqrt{20+x} = 5\)

### Problem 3

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

86

81

9

The equation has no solution.

### Problem 4

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

\(\text- \frac{5}{\sqrt2}\)

\(\frac{1}{\sqrt{2^5}}\)

\(\frac{1}{\sqrt{32}}\)

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

\(\text- 2^{\frac52}\)

\(2^{\text- \frac52}\)

### Problem 5

Which are the solutions to the equation \(x^2=36\)?

6 only

-6 only

6 and -6

This equation has no solutions.

### Problem 6

Here is a graph of \(y=x^2\).

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

### 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.