Question

solve the following quadratic equation using square roots. write your answer in set notation.\\(\frac{(x + 27)^2}{-6} = -24\\)\\(\\) (1 point)\\(\\{\square\\}\\)

Explanation:

Step1: Multiply both sides by -6

To eliminate the denominator, we multiply both sides of the equation \(\frac{(x + 27)^2}{-6}=-24\) by \(-6\).
\(\frac{(x + 27)^2}{-6}\times(-6)=-24\times(-6)\)
Simplifying both sides, we get \((x + 27)^2 = 144\)

Step2: Take square roots on both sides

Taking the square root of both sides, we have \(x + 27=\pm\sqrt{144}\)
Since \(\sqrt{144} = 12\), this becomes \(x + 27=\pm12\)

Step3: Solve for x in two cases

Case 1: When \(x + 27 = 12\)

Subtract 27 from both sides: \(x=12 - 27=-15\)

Case 2: When \(x + 27=-12\)

Subtract 27 from both sides: \(x=-12 - 27=-39\)

Answer:

\(\{-39, -15\}\)