Question

Use the table to answer the 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

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 2

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 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**.