Question

solve.

\\\sqrt{2x + 7} = x - 4\\

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

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

Explanation:

🆕 New Concept Discovered: Solving Radical Equations
Eliminate the root, solve, and check for fake solutions.

Step 1: Square both sides

To clear the square root, square both sides of the equation:

$$ (\sqrt{2x + 7})^2 = (x - 4)^2 $$

Simplify both sides:

$$ 2x + 7 = x^2 - 8x + 16 $$

Step 2: Set to zero and solve

Move all terms to one side to form a quadratic equation:

$$ x^2 - 10x + 9 = 0 $$

Factor the quadratic expression:

$$ (x - 9)(x - 1) = 0 $$

This gives two potential solutions:

$$ x = 9 \quad \text{or} \quad x = 1 $$

Step 3: Check for extraneous solutions

We must plug both values back into the original equation \(\sqrt{2x + 7} = x - 4\) to verify them, because squaring can introduce fake (extraneous) solutions.

For \(x = 9\):

$$ \sqrt{2(9) + 7} = 9 - 4 $$

$$ \sqrt{25} = 5 $$

$$ 5 = 5 \quad \text{(True)} $$

For \(x = 1\):

$$ \sqrt{2(1) + 7} = 1 - 4 $$

$$ \sqrt{9} = -3 $$

$$ 3 = -3 \quad \text{(False)} $$

Thus, \(x = 1\) is an extraneous solution and must be discarded.

Answer:

A. The solution set is {9}.