Question

what are the solutions to the quadratic equation ((5y + 6)^2 = 24)?
y = \frac{4\sqrt{6}}{5} and y = \frac{8\sqrt{6}}{5}
y = \frac{-6 + 2\sqrt{6}}{5} and y = \frac{6 - 2\sqrt{6}}{5}
y = \frac{-4\sqrt{6}}{5} and y = \frac{-8\sqrt{6}}{5}

Explanation:

Step1: Take square roots of both sides

$5y + 6 = \pm\sqrt{24}$
Simplify $\sqrt{24}$ to $2\sqrt{6}$, so $5y + 6 = \pm2\sqrt{6}$

Step2: Isolate the term with $y$

$5y = -6 \pm 2\sqrt{6}$

Step3: Solve for $y$

$y = \frac{-6 \pm 2\sqrt{6}}{5}$
This splits into two solutions: $y = \frac{-6+2\sqrt{6}}{5}$ and $y = \frac{-6-2\sqrt{6}}{5}$ (note: the second option's second solution has a sign typo, the correct second solution is $\frac{-6-2\sqrt{6}}{5}$, but among the given choices, the matching pair is the second option)

Answer:

$y = \frac{-6+2\sqrt{6}}{5}$ and $y = \frac{6-2\sqrt{6}}{5}$ (corrected second solution should be $\frac{-6-2\sqrt{6}}{5}$, this is the closest valid option provided)