https://file.aiquickdraw.com/mathbot/file/1725679884155uamvq44k.jpg

The problem in the image asks you to solve the following equation for \( x \):

\[
14x - 5\sqrt{9x + 6} = 42
\]

Let’s solve this step by step.

### Step 1: Isolate the square root term
Move all terms except the square root to one side of the equation:

\[
5\sqrt{9x + 6} = 14x - 42
\]

### Step 2: Square both sides
To eliminate the square root, square both sides of the equation:

\[
\left(5\sqrt{9x + 6}\right)^2 = (14x - 42)^2
\]

This simplifies to:

\[
25(9x + 6) = (14x - 42)^2
\]

### Step 3: Expand both sides
First, expand the terms on both sides:

\[
25(9x + 6) = 25 \cdot 9x + 25 \cdot 6 = 225x + 150
\]

Next, expand \( (14x - 42)^2 \):

\[
(14x - 42)^2 = (14x)^2 - 2 \cdot 14x \cdot 42 + (-42)^2 = 196x^2 - 1176x + 1764
\]

### Step 4: Combine into one equation
Now, we have:

\[
225x + 150 = 196x^2 - 1176x + 1764
\]

### Step 5: Rearrange into standard quadratic form
Bring all terms to one side:

\[
0 = 196x^2 - 1176x + 1764 - 225x - 150
\]

Simplify the linear and constant terms:

\[
0 = 196x^2 - 1401x + 1614
\]

### Step 6: Solve the quadratic equation
Use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 196x^2 - 1401x + 1614 = 0 \), the coefficients are:
- \( a = 196 \)
- \( b = -1401 \)
- \( c = 1614 \)

Let’s calculate the values of \( x \). I’ll do the computation

Solve the equation 14x - 5√(9x + 6) = 42 for x.

This step-by-step guide explains how to solve the equation 14x - 5√(9x + 6) = 42, involving radical expressions and quadratic equations. This problem introduces key algebraic concepts suitable for high school students in Grades 9-12.

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