Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x+27)^2/−6=−3 .(1 point)

Answer 1

To solve the equation \(\frac{(x + 27)^2}{-6} = -3\), we first eliminate the fraction by multiplying both sides by \(-6\):

\[ (x + 27)^2 = -3 \times -6 \]

This simplifies to:

\[ (x + 27)^2 = 18 \]

Next, we take the square root of both sides. Remembering to include both the positive and negative roots, we have:

\[ x + 27 = \sqrt{18} \quad \text{or} \quad x + 27 = -\sqrt{18} \]

Calculating \(\sqrt{18}\):

\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \approx 4.24 \]

Now, we can simplify our two equations:

1. \(x + 27 = 3\sqrt{2}\)
2. \(x + 27 = -3\sqrt{2}\)

For the first equation:

\[ x = 3\sqrt{2} - 27 \approx 4.24 - 27 \approx -22.76 \]

For the second equation:

\[ x = -3\sqrt{2} - 27 \approx -4.24 - 27 \approx -31.24 \]

Thus, the solutions to the quadratic equation are approximately:

\[ x \approx -22.76 \quad \text{and} \quad x \approx -31.24 \]

So, the final rounded answers are:

\[ \boxed{-22.76} \quad \text{and} \quad \boxed{-31.24} \]

Answer 2

why 2? I can only put 1 idiot

Answer 3

I apologize for the misunderstanding. You are looking for a single solution. Since the original equation resulted in two possible values for \(x\), it suggests you were looking for the principal solution.

Let’s summarize clearly, focusing on how to present just one response based on the context of solving the equation.

For the equation:

\[ \frac{(x + 27)^2}{-6} = -3 \]

We simplified it to:

\[ (x + 27)^2 = 18 \]

Taking the square root, we had:

\[ x + 27 = \sqrt{18} \quad \text{or} \quad x + 27 = -\sqrt{18} \]

Calculating \(3\sqrt{2} \approx 4.24\), we found:

1. Positive solution: \[ x = 3\sqrt{2} - 27 \approx -22.76 \]

2. Negative solution: \[ x = -3\sqrt{2} - 27 \approx -31.24 \]

If you need just one of these solutions, typically, the positive branch is often preferred unless specified otherwise. In that case, the rounded answer would be:

\[ \boxed{-22.76} \]

If you'd like the negative solution, that would be \(-31.24\).

Please confirm your preferences for which solution you need, and I can help accordingly! Thank you for your understanding.