https://file.aiquickdraw.com/mathbot/file/1726502643851g4qgzckw.jpg

The equation provided in the image is:

\[
(x - 6)^2 = 51
\]

### Steps to solve:

1. **Take the square root of both sides** to eliminate the square:
   \[
   \sqrt{(x - 6)^2} = \sqrt{51}
   \]
   This simplifies to:
   \[
   x - 6 = \pm \sqrt{51}
   \]
   
2. **Simplify the square root of 51**. Since 51 is not a perfect square, approximate it:
   \[
   \sqrt{51} \approx 7.141
   \]

3. **Solve for \(x\)** by adding 6 to both sides:
   \[
   x = 6 \pm 7.141
   \]

   This gives two possible solutions:
   \[
   x = 6 + 7.141 = 13.141
   \]
   and
   \[
   x = 6 - 7.141 = -1.141
   \]

### Final Answers (rounded to the nearest hundredth):
\[
x \approx 13.14 \quad \text{or} \quad x \approx -1.14
\]

Would you like further details or explanations on any of the steps?

Here are 5 related questions to expand on this topic:
1. How do you solve other types of quadratic equations using the quadratic formula?
2. What are the properties of square roots and how are they applied in different types of problems?
3. How do you solve quadratic equations that cannot be factored?
4. What are complex numbers and how are they related to square roots of negative numbers?
5. How do you solve equations involving cube roots or higher-order roots?

**Tip:** When dealing with square roots, always remember to include both the positive and negative roots in your solutions.

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

Learn how to solve the quadratic equation (x - 6)^2 = 51 using square roots. This step-by-step guide explains the process of taking the square root, solving for x, and rounding to the nearest hundredth. Suitable for Grades 8-10.

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