Question

solve the quadratic equation by the square root method and write the solutions in radical form. simplify the solutions.
$(5y + 13)^2 = 20$
answer 2 points
$y = \square$ or $\square$

Explanation:

Step1: Take square root of both sides

Take the square root of each side of the equation \((5y + 13)^2 = 20\). Remember that when we take the square root of a square, we get both the positive and negative roots. So we have \(5y + 13=\pm\sqrt{20}\).

Step2: Simplify the radical

Simplify \(\sqrt{20}\). We know that \(20 = 4\times5\), and \(\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5}\). So the equation becomes \(5y + 13=\pm2\sqrt{5}\).

Step3: Solve for y

Subtract 13 from both sides: \(5y=- 13\pm2\sqrt{5}\). Then divide both sides by 5: \(y=\frac{-13\pm2\sqrt{5}}{5}\). We can split this into two solutions: \(y = \frac{-13 + 2\sqrt{5}}{5}\) and \(y=\frac{-13-2\sqrt{5}}{5}\).

Answer:

\(y=\frac{-13 + 2\sqrt{5}}{5}\) or \(y=\frac{-13-2\sqrt{5}}{5}\)