https://file.aiquickdraw.com/mathbot/file/1725874863298pmtm8oia.jpg

The image contains the following equation to be solved:

\[
\frac{6}{\sqrt{x-1}} - 2\sqrt{x-1} = 1
\]

Let's solve this step-by-step.

### Step 1: Substitute \( y = \sqrt{x - 1} \)
This substitution simplifies the equation. Hence, the equation becomes:
\[
\frac{6}{y} - 2y = 1
\]

### Step 2: Eliminate the fraction
Multiply the entire equation by \( y \) to eliminate the fraction:
\[
6 - 2y^2 = y
\]

### Step 3: Rearrange the equation
Move all terms to one side to form a quadratic equation:
\[
2y^2 + y - 6 = 0
\]

### Step 4: Solve the quadratic equation
We'll use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 1 \), and \( c = -6 \).

\[
y = \frac{-1 \pm \sqrt{1^2 - 4(2)(-6)}}{2(2)}
\]
\[
y = \frac{-1 \pm \sqrt{1 + 48}}{4}
\]
\[
y = \frac{-1 \pm \sqrt{49}}{4}
\]
\[
y = \frac{-1 \pm 7}{4}
\]

Thus, \( y \) has two possible values:
\[
y = \frac{-1 + 7}{4} = \frac{6}{4} = 1.5
\]
or
\[
y = \frac{-1 - 7}{4} = \frac{-8}{4} = -2
\]

### Step 5: Back-substitute \( y = \sqrt{x - 1} \)
Now, solve for \( x \):

- If \( y = 1.5 \):
  \[
  \sqrt{x - 1} = 1.5 \implies x - 1 = 2.25 \implies x = 3.25
  \]

- If \( y = -2 \), this does not give a valid solution since square roots are non-negative.

### Final Solution:
The only valid solution is \( x = 3.25 \).

Would you like further details or explanations?

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Here are some related questions for further practice:
1. How do you solve quadratic equations using the quadratic formula?
2. What is the purpose of substituting variables in solving equations with square roots?
3. How can you check if a solution to a radical equation is valid?
4. What are common mistakes when solving equations involving fractions and radicals?
5. How does squaring both sides of an equation help in solving radical equations?

**Tip:** Always check for extraneous solutions when solving radical equations, as squaring both sides can introduce invalid results.

Solve the equation: \(\frac{6}{\sqrt{x-1}} - 2\sqrt{x-1} = 1\)

Learn how to solve a radical equation step-by-step: \(\frac{6}{\sqrt{x-1}} - 2\sqrt{x-1} = 1\). This guide covers algebraic concepts such as substitution, solving quadratic equations, and applying the quadratic formula. Suitable for students in Grades 9-12.

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