For the quadratic equation (x−7)2=11

Question

, 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 solve the quadratic equation \((x - 7)^2 = 11\), we first take the square root of both sides:

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

This gives us two equations to solve:

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

For the first equation:

\[ x = 7 + \sqrt{11} \]

For the second equation:

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

Now, we can determine the approximate values of these solutions:

1. The approximate value of \( \sqrt{11} \) is about 3.32.

   • So for \(x = 7 + \sqrt{11}\): \[ x \approx 7 + 3.32 = 10.32 \]

2. And for \(x = 7 - \sqrt{11}\): \[ x \approx 7 - 3.32 = 3.68 \]

Now we can identify the two solutions:

• The smaller of the two solutions is \(x \approx 3.68\).
• The larger of the two solutions is \(x \approx 10.32\).

Finally, we can fill in the answer:

The smaller of the two solutions is \(x = 7 - \sqrt{11} \approx 3.68\). The larger of the two solutions is \(x = 7 + \sqrt{11} \approx 10.32\).

Thus:

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