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 find the solutions to the quadratic equation \((x−7)^2=11\), we first need to solve the equation.

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

2. This gives us two separate equations to solve:

   • \(x - 7 = \sqrt{11}\)
   • \(x - 7 = -\sqrt{11}\)

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 can identify the two solutions:

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

To approximate these values, we can use \(\sqrt{11} \approx 3.3166\):

• Smaller solution: \(x \approx 7 - 3.3166 \approx 3.6834\)
• Larger solution: \(x \approx 7 + 3.3166 \approx 10.3166\)

Thus, we summarize the solutions:

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

So the final answers are:

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