Question

solve the equation.

\\\sqrt{74 - 10x} + 5 = x\\

select the correct choice below and fill in any answer boxes in your choice.

a. the solution set is \\{ \\}.
(type an integer or a fraction. use a comma to separate answers as needed.)
b. the solution set is the empty set.

Explanation:

🆕 New Concept Discovered: Solving Radical Equations
Isolate the radical, square both sides, and check for extraneous solutions.

Step 1: Isolate the radical term

Subtract \( 5 \) from both sides of the equation to get the square root term by itself:

$$ \sqrt{74 - 10x} = x - 5 $$

Step 2: Square both sides

Eliminate the radical by squaring both sides of the equation:

$$ (\sqrt{74 - 10x})^2 = (x - 5)^2 $$

$$ 74 - 10x = x^2 - 10x + 25 $$

Step 3: Solve the quadratic equation

Add \( 10x \) and subtract \( 74 \) from both sides to set the quadratic equation to zero:

$$ 0 = x^2 - 10x + 10x + 25 - 74 $$

$$ x^2 - 49 = 0 $$

Factor the difference of squares:

$$ (x - 7)(x + 7) = 0 $$

This gives two potential solutions:

$$ x = 7 \quad \text{or} \quad x = -7 $$

Step 4: Check for extraneous solutions

We must substitute both potential solutions back into the original equation, \( \sqrt{74 - 10x} + 5 = x \), to verify them.

Test \( x = 7 \):

$$ \sqrt{74 - 10(7)} + 5 = 7 $$

$$ \sqrt{74 - 70} + 5 = 7 $$

$$ \sqrt{4} + 5 = 7 $$

$$ 2 + 5 = 7 \quad \text{(True)} $$

Test \( x = -7 \):

$$ \sqrt{74 - 10(-7)} + 5 = -7 $$

$$ \sqrt{74 + 70} + 5 = -7 $$

$$ \sqrt{144} + 5 = -7 $$

$$ 12 + 5 = 17 eq -7 \quad \text{(False)} $$

Thus, \( x = -7 \) is an extraneous solution.

Answer:

A. The solution set is \( \{7\} \).