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

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

Then, we can solve for \(x\):

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

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

Now we have the two solutions:

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

To get numerical approximations for the roots:
- \(\sqrt{11} \approx 3.32\)
- Therefore:
- The smaller solution is approximately \(7 - 3.32 \approx 3.68\).
- The larger solution is approximately \(7 + 3.32 \approx 10.32\).

Thus, the two solutions to the quadratic equation are:

- The smaller of the two solutions is \(x \approx 3.68\) or the exact form \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x \approx 10.32\) or the exact form \(x = 7 + \sqrt{11}\).

Answer:

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