Question

Use the table to answer the question.

x Substituted Evaluate True Statement?
7−11−−√

7
11−7–√

7+11−−√

11+7–√

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=
.
Remaining Attempts : 3
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page 15 of 16

Answer 1

To solve the quadratic equation \((x - 7)^2 = 11\), we'll first find the solutions by taking the square root of both sides.

1. Start by rewriting the equation:
\[(x - 7)^2 = 11\]
2. Take the square root of both sides:
\[x - 7 = \pm \sqrt{11}\]

This leads to two equations:
\[x - 7 = \sqrt{11}\]
\[x - 7 = -\sqrt{11}\]

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 have the two solutions:
- The smaller solution is \(x = 7 - \sqrt{11}\)
- The larger solution is \(x = 7 + \sqrt{11}\)

We can approximate \(\sqrt{11}\):
\(\sqrt{11} \approx 3.3166\)

Thus:
- The smaller solution:
\[x \approx 7 - 3.3166 \approx 3.6834\]
- The larger solution:
\[x \approx 7 + 3.3166 \approx 10.3166\]

However, for exact values:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).

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