Question

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 quadratic equation \((x + 27)^2 - 6 = -3\), we'll follow these steps:

1. **Isolate the squared term**: Start by adding 6 to both sides of the equation.

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

2. **Take the square root of both sides**: When taking the square root, remember to include both the positive and negative roots.

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

3. **Solve for \(x\)**: Subtract 27 from both sides for both cases.

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

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

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

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

4. **Calculate the numerical values**: Using the approximation \(\sqrt{3} \approx 1.732\):

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

\[
x \approx 1.732 - 27 \approx -25.268
\]

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

\[
x \approx -1.732 - 27 \approx -28.732
\]

5. **Round the results to the nearest hundredth**:

- First solution: \(x \approx -25.27\)

- Second solution: \(x \approx -28.73\)

Thus, the solutions to the quadratic equation are:

\[
x \approx -25.27 \quad \text{and} \quad x \approx -28.73
\]