Question

Solve for
\[x\]. Enter the solutions from least to greatest.
Round to two decimal places.
\[(x + 8)^2 - 2 = 0\]
\[\text{lesser }x = \]
\[\text{greater } x = \]

Answer 1

To solve the equation

\[
(x + 8)^2 - 2 = 0,
\]

we first isolate the squared term:

\[
(x + 8)^2 = 2.
\]

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

\[
x + 8 = \sqrt{2} \quad \text{or} \quad x + 8 = -\sqrt{2}.
\]

Now we will solve for \(x\) in each case.

1. For the first case:

\[
x + 8 = \sqrt{2}.
\]
Subtracting 8 from both sides gives:

\[
x = \sqrt{2} - 8.
\]

2. For the second case:

\[
x + 8 = -\sqrt{2}.
\]
Again, subtracting 8 from both sides gives:

\[
x = -\sqrt{2} - 8.
\]

Next, we calculate the numerical values of these solutions.

We know approximately:

\[
\sqrt{2} \approx 1.414.
\]

Thus, we can compute:

1. For \(x = \sqrt{2} - 8\):

\[
x \approx 1.414 - 8 \approx -6.586.
\]

2. For \(x = -\sqrt{2} - 8\):

\[
x \approx -1.414 - 8 \approx -9.414.
\]

Now we have the solutions rounded to two decimal places:

- Lesser \(x\): \(-9.41\)
- Greater \(x\): \(-6.59\)

The final answers are:

\[
\text{lesser } x = -9.41
\]

\[
\text{greater } x = -6.59
\]