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. (1 point) The smaller of the two solutions is x= . The larger of the two solutions is x= .

Answer 1

To find the solutions to the quadratic equation \((x−7)^2=11\), we first need to solve the equation.

1. Take the square root of both sides:
\[
x - 7 = \pm \sqrt{11}
\]

2. This gives us two separate equations to solve:

- \(x - 7 = \sqrt{11}\)
- \(x - 7 = -\sqrt{11}\)

3. Solve for \(x\) in each case:

- For \(x - 7 = \sqrt{11}\):
\[
x = 7 + \sqrt{11}
\]

- For \(x - 7 = -\sqrt{11}\):
\[
x = 7 - \sqrt{11}
\]

Now we can identify the two solutions:
- The smaller solution is \(x = 7 - \sqrt{11}\).
- The larger solution is \(x = 7 + \sqrt{11}\).

To approximate these values, we can use \(\sqrt{11} \approx 3.3166\):
- Smaller solution: \(x \approx 7 - 3.3166 \approx 3.6834\)
- Larger solution: \(x \approx 7 + 3.3166 \approx 10.3166\)

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

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