Question

Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x-6)^2 =51.

Answer 1

To solve the equation \((x-6)^2 = 51\) using square roots, follow these steps:

1. **Take the square root of both sides:**
\[
\sqrt{(x-6)^2} = \sqrt{51}
\]
This gives us two equations:
\[
x - 6 = \sqrt{51} \quad \text{or} \quad x - 6 = -\sqrt{51}
\]

2. **Solve for \(x\) in each case:**

For the first equation:
\[
x - 6 = \sqrt{51}
\]
\[
x = \sqrt{51} + 6
\]

For the second equation:
\[
x - 6 = -\sqrt{51}
\]
\[
x = -\sqrt{51} + 6
\]

3. **Calculate \(\sqrt{51}\):**
\(\sqrt{51} \approx 7.1414...\)

4. **Substituting back the approximate value:**

For \(x = \sqrt{51} + 6\):
\[
x \approx 7.1414 + 6 \approx 13.1414
\]

For \(x = -\sqrt{51} + 6\):
\[
x \approx -7.1414 + 6 \approx -1.1414
\]

5. **Final answers rounded to the nearest hundredth:**
\[
x \approx 13.14
\]
\[
x \approx -1.14
\]

Thus, the solutions to the equation \((x-6)^2 = 51\) are:

\[
x \approx 13.14 \quad \text{and} \quad x \approx -1.14
\]