Question

find the solution to the equation, or explain why there is no solution. remember to check for extraneous solutions. \\(\sqrt{2x + 10} = \sqrt{x + 7}\\)

Explanation:

Step1: Square both sides to eliminate radicals

To solve the equation \(\sqrt{2x + 10}=\sqrt{x + 7}\), we start by squaring both sides. Squaring a square root cancels out the square root, so we get:

$$ (\sqrt{2x + 10})^2=(\sqrt{x + 7})^2 $$

Simplifying both sides, we have:

$$ 2x + 10=x + 7 $$

Step2: Solve for \(x\)

Subtract \(x\) from both sides:

$$ 2x - x+ 10=x - x+ 7 $$

Which simplifies to:

$$ x + 10=7 $$

Then subtract 10 from both sides:

$$ x + 10-10=7 - 10 $$

So, \(x=- 3\)

Step3: Check for extraneous solutions

We need to check if \(x =-3\) makes the original equation valid. Substitute \(x=-3\) into the original equation:
Left - hand side: \(\sqrt{2(-3)+10}=\sqrt{-6 + 10}=\sqrt{4}=2\)
Right - hand side: \(\sqrt{-3 + 7}=\sqrt{4}=2\)
Since both sides are equal when \(x=-3\), \(x =-3\) is a valid solution.

Answer:

\(x=-3\)