Use the table to answer the question.

Question

x Substituted Evaluate True Statement?
7−11−−√

7
11−7–√

7+11−−√

11+7–√

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 solve the quadratic equation \( (x - 7)^2 = 11 \), we first need to find the values of \( x \) that satisfy the equation.

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

2. Solve for \( x \):

   • For the first equation: \[ x - 7 = \sqrt{11} \Rightarrow x = 7 + \sqrt{11} \]
   • For the second equation: \[ x - 7 = -\sqrt{11} \Rightarrow x = 7 - \sqrt{11} \]

3. The two solutions to the quadratic equation are:

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

To approximate:

• \( \sqrt{11} \approx 3.317 \)
• So, the smaller solution: \[ x \approx 7 - 3.317 \approx 3.683 \]
• And the larger solution: \[ x \approx 7 + 3.317 \approx 10.317 \]

Thus, you can fill in the blanks with:

• The smaller of the two solutions is \( x \approx 3.683 \).
• The larger of the two solutions is \( x \approx 10.317 \).

For the exact representation:

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

Answer 2

How many solutions would the equation x2=−9
have? You do not need to solve for x.(1 point)
The equation would have
solution(s).
Remaining Attempts : 3

Answer 3

The equation \( x^2 = -9 \) has no real solutions because the square of a real number cannot be negative.

However, if we consider complex numbers, we can find solutions. The solutions would be \( x = 3i \) and \( x = -3i \), where \( i \) is the imaginary unit.

Therefore, the equation would have 2 complex solutions.