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:

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