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Use the table to answer the 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=□ .

1 answer

To solve the quadratic equation \((x-7)^2 = 11\), we first need to find the solutions by taking the square root of both sides. Here are the steps:

  1. Rewrite the equation: \[ (x-7)^2 = 11 \]

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

  3. Solve for \(x\): \[ x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11} \]

Now we'll compute the numeric values of each solution.

  • For \(x = 7 + \sqrt{11}\):

    • \(\sqrt{11} \approx 3.317\) (approximately)
    • Thus, \(x \approx 7 + 3.317 = 10.317\)

  • For \(x = 7 - \sqrt{11}\):

    • Thus, \(x \approx 7 - 3.317 = 3.683\)

Now we can state the two solutions:

  • The smaller solution is: \[ x \approx 3.683 \]

  • The larger solution is: \[ x \approx 10.317 \]

In conclusion:

  • The smaller of the two solutions is \(x = 7 - \sqrt{11} \approx 3.683\).
  • The larger of the two solutions is \(x = 7 + \sqrt{11} \approx 10.317\).

Final answers:

  • The smaller of the two solutions is \(x = 7 - \sqrt{11} \approx 3.683\).
  • The larger of the two solutions is \(x = 7 + \sqrt{11} \approx 10.317\).