Question

use the table to answer the question.
evaluating substituted values of x
x\tsubstituted\tevaluate\ttrue statement?
$7 - \sqrt{11}$
7
$11 - \sqrt{7}$
$7 + \sqrt{11}$
$11 + \sqrt{7}$
for the quadratic equation $(x - 7)^2 = 11$, complete the table by inserting each of the given values to determine whether it is a solution. identify the two solutions to the quadratic equation.
note: the numbers in the table are arranged from the smallest (closest to zero) to the largest (farthest from zero).
(1 point)

Explanation:

To solve the quadratic equation \((x - 7)^2 = 11\), we can use the square - root property. If \(y^{2}=a\) (\(a\geq0\)), then \(y=\pm\sqrt{a}\).

Step 1: Solve the equation \((x - 7)^2 = 11\) for \(x\)

Take the square root of both sides of the equation \((x - 7)^2 = 11\). We get \(x - 7=\pm\sqrt{11}\).
Then, solve for \(x\) by adding 7 to both sides of the equation. So \(x = 7\pm\sqrt{11}\), which means \(x = 7+\sqrt{11}\) or \(x = 7 - \sqrt{11}\).

Step 2: Evaluate each value of \(x\) in the table

• For \(x = 7-\sqrt{11}\):

Substitute \(x = 7-\sqrt{11}\) into the left - hand side of the equation \((x - 7)^2\). We have \(( (7-\sqrt{11})-7)^2=(-\sqrt{11})^2 = 11\), which is equal to the right - hand side of the equation \((x - 7)^2 = 11\). So the statement is true.

• For \(x = 7\):

Substitute \(x = 7\) into the left - hand side of the equation \((x - 7)^2\). We get \((7 - 7)^2=0^2 = 0
eq11\). So the statement is false.

• For \(x = 11-\sqrt{7}\):

Substitute \(x = 11-\sqrt{7}\) into the left - hand side of the equation \((x - 7)^2\). We have \(( (11-\sqrt{7})-7)^2=(4-\sqrt{7})^2=16-8\sqrt{7}+7=23 - 8\sqrt{7}\approx23-8\times2.646 = 23 - 21.168 = 1.832
eq11\). So the statement is false.

• For \(x = 11-\sqrt{7}\):

As calculated above, \((x - 7)^2=(4-\sqrt{7})^2
eq11\), so the statement is false.

• For \(x = 7+\sqrt{11}\):

Substitute \(x = 7+\sqrt{11}\) into the left - hand side of the equation \((x - 7)^2\). We have \(( (7+\sqrt{11})-7)^2=(\sqrt{11})^2 = 11\), which is equal to the right - hand side of the equation \((x - 7)^2 = 11\). So the statement is true.

• For \(x = 11+\sqrt{7}\):

Substitute \(x = 11+\sqrt{7}\) into the left - hand side of the equation \((x - 7)^2\). We have \(( (11+\sqrt{7})-7)^2=(4+\sqrt{7})^2=16 + 8\sqrt{7}+7=23+8\sqrt{7}\approx23 + 8\times2.646=23 + 21.168 = 44.168
eq11\). So the statement is false.

The two solutions of the quadratic equation \((x - 7)^2 = 11\) are \(x = 7+\sqrt{11}\) and \(x = 7-\sqrt{11}\)

Completing the table (summary)

\(x\)	Substituted	Evaluate	True Statement?
\(7\)	\((7 - 7)^2\)	\(0\)	No
\(11-\sqrt{7}\)	\(( (11-\sqrt{7})-7)^2\)	\(23 - 8\sqrt{7}\)	No
\(7+\sqrt{11}\)	\(( (7+\sqrt{11})-7)^2\)	\(11\)	Yes
\(11+\sqrt{7}\)	\(( (11+\sqrt{7})-7)^2\)	\(23 + 8\sqrt{7}\)	No

The two solutions of the quadratic equation \((x - 7)^2=11\) are \(x = 7+\sqrt{11}\) and \(x = 7-\sqrt{11}\)

Answer:

The two solutions to the quadratic equation \((x - 7)^2 = 11\) are \(x = 7+\sqrt{11}\) and \(x = 7-\sqrt{11}\)