The given quadratic equation is:

$(x - 7)^2 = 11$

Step 1: Solve the equation algebraically.

To solve this, first take the square root of both sides:

$\sqrt{(x - 7)^2} = \pm \sqrt{11}$

This simplifies to:

$x - 7 = \pm \sqrt{11}$

Now, solve for $x$:

$x = 7 \pm \sqrt{11}$

Step 2: Find the two solutions.

1. First solution (positive root):

$x = 7 + \sqrt{11} \approx 7 + 3.317 = 10.317$

2. Second solution (negative root):

$x = 7 - \sqrt{11} \approx 7 - 3.317 = 3.683$

Final Answer:

• The smaller of the two solutions is $x \approx 3.683$.
• The larger of the two solutions is $x \approx 10.317$.

Do you want further details or have any questions?

Relative Questions:

1. How does taking the square root of both sides help solve a quadratic equation?
2. What are other methods for solving quadratic equations, such as factoring or the quadratic formula?
3. Why do we add and subtract the square root when solving the equation?
4. What is the significance of the plus-minus ($\pm$) symbol in solving square root equations?
5. How do you calculate decimal approximations for square roots without a calculator?

Tip:

When solving quadratic equations, remember to check for both the positive and negative roots when taking square roots of both sides.