Question

Solve the fallowing quadratic equation using square roots. Round to the nearest hundredth if necessary.(x+27)^2/-6=-3

Answer 1

To solve the quadratic equation \(\frac{(x + 27)^2}{-6} = -3\), we start by eliminating the fraction. Multiply both sides by \(-6\):

\[
(x + 27)^2 = -3 \cdot -6
\]
\[
(x + 27)^2 = 18
\]

Next, we take the square root of both sides. Remember to consider both the positive and negative square roots:

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

Now, simplify \(\sqrt{18}\):

\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]

So, we have:

\[
x + 27 = 3\sqrt{2} \quad \text{or} \quad x + 27 = -3\sqrt{2}
\]

Now, solve for \(x\) in both cases:

1. **First case:**
\[
x = 3\sqrt{2} - 27
\]

2. **Second case:**
\[
x = -3\sqrt{2} - 27
\]

Now, calculate \(3\sqrt{2}\) (using \(\sqrt{2} \approx 1.414\)):

\[
3\sqrt{2} \approx 3 \cdot 1.414 \approx 4.242
\]

Therefore, for the first case:

\[
x \approx 4.242 - 27 \approx -22.758
\]

For the second case:

\[
-3\sqrt{2} \approx -4.242
\]
\[
x \approx -4.242 - 27 \approx -31.242
\]

Thus, the two solutions are:

\[
x \approx -22.76 \quad \text{(rounded to the nearest hundredth)}
\]
\[
x \approx -31.24 \quad \text{(rounded to the nearest hundredth)}
\]

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