Question

solve for ( x ):
( sqrt{2x + 6} = sqrt{6x - 34} )
( x = )
express all answers as exact integers or reduced fractions--do not round. if there is more than one solution, separate them with a comma. for example, if ( x = -1 ) or ( x = 2 ), write ( -1, 2 ). if there are no solutions, write ( dne ).

Explanation:

Step1: Square both sides to eliminate square roots

To solve the equation \(\sqrt{2x + 6}=\sqrt{6x - 34}\), we can square both sides of the equation. Squaring a square root cancels out the square root, so we get:
\((\sqrt{2x + 6})^2=(\sqrt{6x - 34})^2\)
Which simplifies to:
\(2x + 6=6x - 34\)

Step2: Solve for \(x\)

First, we want to get all the \(x\) terms on one side and the constant terms on the other side. Subtract \(2x\) from both sides:
\(2x+6 - 2x=6x - 34- 2x\)
Simplifying gives:
\(6 = 4x-34\)
Then, add 34 to both sides:
\(6 + 34=4x-34 + 34\)
Simplifying the left and right sides:
\(40 = 4x\)
Now, divide both sides by 4:
\(\frac{40}{4}=\frac{4x}{4}\)
Which gives:
\(x = 10\)

Step3: Check the solution

We need to check if \(x = 10\) satisfies the original equation because squaring both sides can sometimes introduce extraneous solutions.
For the left - hand side: \(\sqrt{2(10)+6}=\sqrt{20 + 6}=\sqrt{26}\)
For the right - hand side: \(\sqrt{6(10)-34}=\sqrt{60 - 34}=\sqrt{26}\)
Since both sides are equal when \(x = 10\), the solution is valid.

Answer:

\(10\)