Question

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

Answer 1

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

1. Take the square root of both sides:
\[
x - 6 = \pm \sqrt{51}
\]

2. Calculate \(\sqrt{51}\):
The approximate value of \(\sqrt{51}\) is about 7.14 (since \(7.14^2 \approx 51\)).

3. Rewrite the equation as two separate equations:
\[
x - 6 = \sqrt{51} \quad \text{and} \quad x - 6 = -\sqrt{51}
\]

4. Solve for \(x\) in both cases:
- For the first equation:
\[
x = 6 + \sqrt{51}
\]
- For the second equation:
\[
x = 6 - \sqrt{51}
\]

5. Now substitute \(\sqrt{51}\) with its approximated value (7.14):
- First solution:
\[
x = 6 + 7.14 \approx 13.14
\]
- Second solution:
\[
x = 6 - 7.14 \approx -1.14
\]

Thus, the two solutions to the equation \((x - 6)^2 = 51\) are:
\[
x \approx 13.14 \quad \text{and} \quad x \approx -1.14
\]

Rounded to the nearest hundredth, the solutions are:
\[
\boxed{13.14} \quad \text{and} \quad \boxed{-1.14}
\]