Question

Use the table to answer the question.%0D%0A%0D%0Ax%09Substituted%09Evaluate%09True Statement?%0D%0A7−11−−√%0D%0A7%0D%0A−%0D%0A11%0D%0A %09 %09 %09 %0D%0A7%09 %09 %09 %0D%0A11−7–√%0D%0A11%0D%0A−%0D%0A7%0D%0A %09 %09 %09 %0D%0A7+11−−√%0D%0A7%0D%0A+%0D%0A11%0D%0A %09 %09 %09 %0D%0A11+7–√%0D%0A11%0D%0A+%0D%0A7%0D%0A %09 %09 %09 %0D%0AFor the quadratic equation (x−7)2=11%0D%0A(%0D%0A𝑥%0D%0A−%0D%0A7%0D%0A)%0D%0A2%0D%0A=%0D%0A11%0D%0A , 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.%0D%0A%0D%0A(1 point)%0D%0AThe smaller of the two solutions is x=%0D%0A𝑥%0D%0A=%0D%0A. The larger of the two solutions is x=%0D%0A𝑥%0D%0A=%0D%0A.

Answer 1

To solve the quadratic equation \((x−7)^2=11\), we will first solve for \(x\) by taking the square root of both sides.

1. Start with the equation: \[ (x−7)^2 = 11 \]

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

3. Solve for \(x\) in both cases:

   a. For \(x−7 = \sqrt{11}\): \[ x = 7 + \sqrt{11} \]

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

Let's provide the approximate numerical values:

• The approximate value of \(\sqrt{11} \approx 3.317\), thus:
  • \( 7 - \sqrt{11} \approx 7 - 3.317 \approx 3.683\)
  • \( 7 + \sqrt{11} \approx 7 + 3.317 \approx 10.317\)

So we identify the two solutions:

• The smaller of the two solutions is: \[ x = 7 - \sqrt{11} \approx 3.683 \]

• The larger of the two solutions is: \[ x = 7 + \sqrt{11} \approx 10.317 \]

Finally, filling in the table with the discovered values:

• The smaller solution is \( x \approx 3.683 \).
• The larger solution is \( x \approx 10.317 \).

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