Question

solve the radical equation.

\\\sqrt{2-x}-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
Isolate the root, square both sides, and check for fake solutions.

Step 1: Isolate the radical term

To clear the square root, we first need to get it by itself on one side of the equation. Add \( x \) to both sides:

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

Step 2: Square both sides

Eliminate the radical by squaring both sides of the equation:

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

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

Step 3: Solve the quadratic equation

Move all terms to one side to set the quadratic equation to zero:

$$ 0 = x^2 + 8x + x + 16 - 2 $$

$$ x^2 + 9x + 14 = 0 $$

Factor the trinomial by finding two numbers that multiply to \( 14 \) and add to \( 9 \) (which are \( 2 \) and \( 7 \)):

$$ (x + 2)(x + 7) = 0 $$

This gives two potential solutions:

$$ x = -2 \quad \text{or} \quad x = -7 $$

Step 4: Check for extraneous solutions

Squaring both sides can introduce "extraneous" (fake) solutions, so we must test both values in the original equation: \( \sqrt{2 - x} - x = 4 \).

• Test \( x = -2 \):

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

This is true, so \( x = -2 \) is a valid solution.

• Test \( x = -7 \):

$$ \sqrt{2 - (-7)} - (-7) = \sqrt{9} + 7 = 3 + 7 = 10 eq 4 $$

This is false, so \( x = -7 \) is an extraneous solution and must be discarded.

Answer:

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