Question

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

Answer 1

To solve the quadratic equation \((x + 27)^2 - 6 = -3\), we start by isolating the square term.

1. **Add 6 to both sides** to eliminate the \(-6\):
\[
(x + 27)^2 - 6 + 6 = -3 + 6
\]
This simplifies to:
\[
(x + 27)^2 = 3
\]

2. **Take the square root of both sides**:
\[
x + 27 = \pm \sqrt{3}
\]

3. **Solve for \(x\)** by isolating it:
\[
x = -27 \pm \sqrt{3}
\]

This gives us two potential solutions:
- First solution:
\[
x = -27 + \sqrt{3}
\]
- Second solution:
\[
x = -27 - \sqrt{3}
\]

4. **Calculate the numerical values**:
- For \(x = -27 + \sqrt{3}\):
\[
\sqrt{3} \approx 1.732
\]
\[
x \approx -27 + 1.732 \approx -25.268
\]
- For \(x = -27 - \sqrt{3}\):
\[
x \approx -27 - 1.732 \approx -28.732
\]

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