We are tasked with solving the equation:

\[
\sqrt{x - 7} = x - 1
\]

We will use trial and error to find values of \(x\) that satisfy this equation. Let's try different values of \(x\) and test them in the equation.

### 1. Try \(x = 5\):
\[
\sqrt{5 - 7} = 5 - 1
\]
\[
\sqrt{-2} = 4
\]
This results in an imaginary number, which means \(x = 5\) is **not** a valid solution.

### 2. Try \(x = 6\):
\[
\sqrt{6 - 7} = 6 - 1
\]
\[
\sqrt{-1} = 5
\]
This also results in an imaginary number, so \(x = 6\) is **not** a valid solution.

### 3. Try \(x = 8\):
\[
\sqrt{8 - 7} = 8 - 1
\]
\[
\sqrt{1} = 7
\]
\[
1 = 7
\]
This is false, so \(x = 8\) is **not** a valid solution.

### 4. Try \(x = 10\):
\[
\sqrt{10 - 7} = 10 - 1
\]
\[
\sqrt{3} = 9
\]
This is also false, so \(x = 10\) is **not** a valid solution.

### 5. Try \(x = 9\):
\[
\sqrt{9 - 7} = 9 - 1
\]
\[
\sqrt{2} = 8
\]
This is false, so \(x = 9\) is **not** a valid solution.

It seems that trial and error may not yield an obvious solution quickly. To solve this algebraically, we can square both sides of the equation.

Would you like to proceed with that method or try more values? If not, we can continue trial and error with different \(x\) values.

---

Here are 5 questions to deepen your understanding:
1. What happens when you square both sides of an equation?
2. Why did the results involve imaginary numbers for some values of \(x\)?
3. What are other ways to approach solving radical equations like this one?
4. How can you check if a solution is valid after squaring both sides?
5. What restrictions do square roots place on possible solutions?

**Tip**: In radical equations, it's important to check for extraneous solutions after squaring both sides.

Explore how to solve the radical equation \sqrt{x - 7} = x - 1 using trial and error. This problem introduces algebraic concepts and explains methods to solve radical equations. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation