Question

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

Enter the smaller of the 2 values first.

(1 point)

Answer 1

To solve the quadratic equation \((x + 27)^2 / -6 = -3\), we start by eliminating the fraction by multiplying both sides by -6:

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

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

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

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

We can simplify \(\sqrt{18}\):

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

Thus, we have:

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

Now, we can solve for \(x\) in each case:

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

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

Calculating \(3\sqrt{2}\):

\[
3\sqrt{2} \approx 3 \times 1.414 = 4.242
\]

Thus:

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

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

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

Now calculating:

\[
-3\sqrt{2} \approx -4.242
\]

Thus:

\[
x \approx -4.242 - 27 = -31.242
\]

Now, rounding both values to the nearest hundredth, we have:

- \(x_1 \approx -22.76\)
- \(x_2 \approx -31.24\)

The smaller value first, we get:

\[
\boxed{-31.24}, \boxed{-22.76}
\]