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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

1 answer

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

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