What are the solutions to the quadratic equation (5y + 6)2 = 24?

y = StartFraction negative 6 + 2 StartRoot 6 EndRoot Over 5 EndFraction and y = StartFraction negative 6 minus 2 StartRoot 6 EndRoot Over 5 EndFraction
y = StartFraction negative 6 + 2 StartRoot 6 EndRoot Over 5 EndFraction and y = StartFraction 6 minus 2 StartRoot 6 EndRoot Over 5 EndFraction
y = StartFraction negative 4 StartRoot 6 EndRoot Over 5 EndFraction and y = StartFraction negative 8 StartRoot 6 EndRoot Over 5 EndFraction
y = StartFraction 4 StartRoot 6 EndRoot Over 5 EndFraction and y = StartFraction 8 StartRoot 6 EndRoot Over 5 EndFraction

1 answer

To solve the quadratic equation \((5y + 6)^2 = 24\), we can follow these steps:

  1. Take the square root of both sides: \[ 5y + 6 = \pm \sqrt{24} \]

    Since \(\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\), we can rewrite this as: \[ 5y + 6 = \pm 2\sqrt{6} \]

  2. Solve for \(y\): We consider both cases:

    • For \(5y + 6 = 2\sqrt{6}\): \[ 5y = 2\sqrt{6} - 6 \] \[ y = \frac{2\sqrt{6} - 6}{5} \]

    • For \(5y + 6 = -2\sqrt{6}\): \[ 5y = -2\sqrt{6} - 6 \] \[ y = \frac{-2\sqrt{6} - 6}{5} \]

  3. Express the solutions: The solutions to the quadratic equation are: \[ y = \frac{2\sqrt{6} - 6}{5} \quad \text{and} \quad y = \frac{-2\sqrt{6} - 6}{5} \]

Now, comparing with the provided answer choices:

  • \(y = \frac{-6 + 2\sqrt{6}}{5}\) and \(y = \frac{-6 - 2\sqrt{6}}{5}\)

These match our derived solutions:

  • The correct answer is: \(y = \frac{-6 + 2\sqrt{6}}{5}\) and \(y = \frac{-6 - 2\sqrt{6}}{5}\)

Hence, it corresponds to the first option: y = StartFraction negative 6 + 2 StartRoot 6 EndRoot Over 5 EndFraction and y = StartFraction negative 6 minus 2 StartRoot 6 EndRoot Over 5 EndFraction.