Question

For the quadratic equation (x - 7)^2 = 11, complete the table by testing each of the given values to determine whether it is a solution. Identify the two solutions to the quadratic equation.

The smaller of the two solutions is x = ?
The larger of the two solutions is x = ?

Answer 1

To solve the quadratic equation \((x - 7)^2 = 11\), we start by taking the square root of both sides:

\[
x - 7 = \pm \sqrt{11}
\]

This gives us two cases to consider:

1. \(x - 7 = \sqrt{11}\)
2. \(x - 7 = -\sqrt{11}\)

### Solving for \(x\):

1. From \(x - 7 = \sqrt{11}\):
\[
x = 7 + \sqrt{11}
\]

2. From \(x - 7 = -\sqrt{11}\):
\[
x = 7 - \sqrt{11}
\]

### Identifying the solutions:

The smaller solution is \(x = 7 - \sqrt{11}\), and the larger solution is \(x = 7 + \sqrt{11}\).

### Approximate values of the solutions:

To approximate these values using \(\sqrt{11} \approx 3.32\):

- Smaller solution:
\[
x = 7 - \sqrt{11} \approx 7 - 3.32 \approx 3.68
\]

- Larger solution:
\[
x = 7 + \sqrt{11} \approx 7 + 3.32 \approx 10.32
\]

### Final Answer:

- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).

In summary:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).