X | Substituted | Evaluate | True Statement?

To determine which values are solutions to the quadratic equation \( z^2 + 20x = 69 \), we need to substitute each given value of \( x \) into the equation and evaluate whether the equation holds true.

First, let's rearrange the equation to express \( z^2 \):

\[
z^2 = 69 - 20x
\]

Now we'll substitute each value of \( x \) into this equation:

1. **For \( x = -23 \)**:
\[
z^2 = 69 - 20(-23) = 69 + 460 = 529
\]
\[
z = \pm \sqrt{529} = \pm 23
\]
This is a valid solution.

2. **For \( x = 0 \)**:
\[
z^2 = 69 - 20(0) = 69
\]
\[
z = \pm \sqrt{69} \quad (\text{This is a real number, but not an integer.})
\]
This is a valid solution but not an integer.

3. **For \( x = 12 \)**:
\[
z^2 = 69 - 20(12) = 69 - 240 = -171
\]
Since \( z^2 \) cannot be negative (in the real number domain), this value does not produce a valid solution.

Now let's fill in the table:

| X | Substituted | Evaluate | True Statement? |
|-----|---------------------------|----------------------|-------------------------------|
| -23 | \( z^2 = 69 - 20(-23) \) | \( z^2 = 529 \) | True (valid solutions: ±23) |
| 0 | \( z^2 = 69 - 20(0) \) | \( z^2 = 69 \) | True (valid solutions: ±√69) |
| 12 | \( z^2 = 69 - 20(12) \) | \( z^2 = -171 \) | False (no real solutions) |

From the evaluation, we conclude:
- The value \( -23 \) produces integer solutions.
- The value \( 0 \) produces real solutions.
- The value \( 12 \) does not produce any real solutions.

Thus, it can be said that:

**Only \( x = -23 \) is in the solution set.**

X | Substituted | Evaluate |True 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 testing each of the given values to determine whether it is a solution. Identify the two solutions to the quadratic equation.
The smaller of the two solutions is x = ?. The larger of the two solutions is x= ?

To solve the quadratic equation \( (x - 7)^2 = 11 \), we will first find the solutions by taking the square root of both sides of the equation:

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

This gives us two equations to solve for \( x \):

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

Now, we can isolate \( x \) in each case:

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

These two values are our potential solutions. Next, we will test the provided values to see if they are equal to either of these solutions.

The possible values to test are:
1. \( 7 - \sqrt{11} \)
2. \( 7 \)
3. \( 11 - \sqrt{7} \)
4. \( 7 + \sqrt{11} \)
5. \( 11 + \sqrt{7} \)

Now let's create the evaluation table.

### Evaluation of Each Value:

1. **For \( x = 7 - \sqrt{11} \)**:
\[
(7 - \sqrt{11} - 7)^2 = (-\sqrt{11})^2 = 11 \quad \text{(True)}
\]

2. **For \( x = 7 \)**:
\[
(7 - 7)^2 = 0 \quad \text{(False)}
\]

3. **For \( x = 11 - \sqrt{7} \)**:
\[
(11 - \sqrt{7} - 7)^2 = (4 - \sqrt{7})^2 \quad \text{(Evaluate)}\\
= (16 - 8\sqrt{7} + 7) = 23 - 8\sqrt{7} \quad \text{(Not equal to 11, False)}
\]

4. **For \( x = 7 + \sqrt{11} \)**:
\[
(7 + \sqrt{11} - 7)^2 = (\sqrt{11})^2 = 11 \quad \text{(True)}
\]

5. **For \( x = 11 + \sqrt{7} \)**:
\[
(11 + \sqrt{7} - 7)^2 = (4 + \sqrt{7})^2\\
= 16 + 8\sqrt{7} + 7 = 23 + 8\sqrt{7} \quad \text{(Not equal to 11, False)}
\]

### Compiling the Evaluation Table:

| X | Substituted | Evaluate | True Statement? |
|----------------------|--------------------------------------|-----------------------------------|-----------------------------|
| \( 7 - \sqrt{11} \) | \( (7 - \sqrt{11} - 7)^2 \) | \( 11 \) | True |
| \( 7 \) | \( (7 - 7)^2 \) | \( 0 \) | False |
| \( 11 - \sqrt{7} \) | \( (11 - \sqrt{7} - 7)^2 \) | Not equal to \( 11 \) | False |
| \( 7 + \sqrt{11} \) | \( (7 + \sqrt{11} - 7)^2 \) | \( 11 \) | True |
| \( 11 + \sqrt{7} \) | \( (11 + \sqrt{7} - 7)^2 \) | Not equal to \( 11 \) | False |

### Conclusion:
The two solutions to the quadratic equation \( (x - 7)^2 = 11 \) are:
- The smaller solution: \( x = 7 - \sqrt{11} \)
- The larger solution: \( x = 7 + \sqrt{11} \)

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

How many solutions would the equation x^2 = -9 have? You do not need to solve for x.

The equation would have ? solution(s).

The equation \( x^2 = -9 \) does not have real solutions because the square of a real number cannot be negative.

However, we can consider complex solutions as well. When we take the square root of both sides, we find:

\[
x = \pm \sqrt{-9} = \pm 3i
\]

Thus, while there are no real solutions, there are two complex solutions: \( 3i \) and \( -3i \).

Therefore, the equation would have **2 solutions** (in the complex plane).

thats wrong

I apologize for any confusion. Let's clarify the situation regarding the equation \( x^2 = -9 \).

### Analysis of the Equation:

The equation \( x^2 = -9 \) implies that we are looking for values of \( x \) that, when squared, yield a negative number. In the set of real numbers, squaring any real number results in a non-negative outcome (i.e., either zero or positive). Therefore, there are **no real solutions** to the equation.

However, in the realm of complex numbers, we can represent the square root of \(-9\) as:

\[
x = \pm \sqrt{-9} = \pm 3i
\]

### Conclusion:

- **Real Solutions:** 0
- **Complex Solutions:** 2 (specifically \( 3i \) and \( -3i \))

Thus, if the question specifically asks for solutions without distinguishing between real and complex:

- The equation \( x^2 = -9 \) has **2 solutions** (considering complex numbers).

If you were looking only for real solutions, the correct statement is that the equation has **0 real solutions**. Thank you for your patience, and I appreciate your opportunity to clarify!