Question

57 solve the equation. check for extraneous solutions. -5 = (x + 7)^(1/2) - 11

Explanation:

Step1: Isolate the radical term

Add 11 to both sides of the equation \(-5=(x + 7)^{\frac{1}{2}}-11\).
We get \(-5 + 11=(x + 7)^{\frac{1}{2}}-11 + 11\), which simplifies to \(6=(x + 7)^{\frac{1}{2}}\).

Step2: Eliminate the fractional exponent

Square both sides of the equation \(6=(x + 7)^{\frac{1}{2}}\) to get rid of the square - root (since \((a^{\frac{1}{2}})^2=a\) for \(a\geq0\)).
\(6^{2}=((x + 7)^{\frac{1}{2}})^{2}\), so \(36=x + 7\).

Step3: Solve for x

Subtract 7 from both sides of the equation \(36=x + 7\).
\(x=36 - 7=29\).

Step4: Check for extraneous solutions

Substitute \(x = 29\) into the original equation \(-5=(x + 7)^{\frac{1}{2}}-11\).
Left - hand side: \(-5\).
Right - hand side: \((29 + 7)^{\frac{1}{2}}-11=\sqrt{36}-11 = 6-11=-5\).
Since the left - hand side equals the right - hand side when \(x = 29\), \(x = 29\) is a valid solution.

Answer:

\(x = 29\)