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.