Question

solve the equation.

\\\sqrt{x + 5} = \sqrt{2x + 3}\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution set is \\{\quad\\}.
(simplify your answer. use a comma to separate answers as needed.)
b. the solution set is the empty set.

Explanation:

🆕 New Concept Discovered: Solving Radical Equations
Eliminate radicals by squaring both sides

Step 1: Square both sides

To eliminate the square roots, square both sides of the equation:

$$ (\sqrt{x + 5})^2 = (\sqrt{2x + 3})^2 $$

$$ x + 5 = 2x + 3 $$

Step 2: Solve for x

Isolate the variable \(x\) by subtracting \(x\) and \(3\) from both sides:

$$ 5 - 3 = 2x - x $$

$$ 2 = x $$

$$ x = 2 $$

Step 3: Verify the solution

Check if \(x = 2\) is a valid solution by plugging it back into the original equation:

$$ \sqrt{2 + 5} = \sqrt{2(2) + 3} $$

$$ \sqrt{7} = \sqrt{4 + 3} $$

$$ \sqrt{7} = \sqrt{7} $$

Since both sides are equal and the values under the radicals are positive, \(x = 2\) is a valid solution.

Answer:

A. The solution set is \(\{2\}\).