Question

x Substituted Evaluate True
-23 ? ? ?
0 ? ? ?
12 ? ? ?
x^2+20x =69

Answer 1

To solve the equation \( x^2 + 20x = 69 \), we need to rearrange it into standard quadratic form:

\[
x^2 + 20x - 69 = 0
\]

Now we will use the quadratic formula to find the values of \( x \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 20 \), and \( c = -69 \).

Calculating the discriminant (\( b^2 - 4ac \)):
\[
b^2 - 4ac = 20^2 - 4 \cdot 1 \cdot (-69) = 400 + 276 = 676
\]

Now we will find the square root of the discriminant:
\[
\sqrt{676} = 26
\]

Now, substituting into the quadratic formula:
\[
x = \frac{-20 \pm 26}{2 \cdot 1}
\]

Calculating the two possible values for \( x \):
1.
\[
x = \frac{-20 + 26}{2} = \frac{6}{2} = 3
\]
2.
\[
x = \frac{-20 - 26}{2} = \frac{-46}{2} = -23
\]

Thus, the solutions are \( x = 3 \) and \( x = -23 \).

Now we’ll fill in the table provided with \( -23 \), \( 0 \), and \( 12 \) substituted into the left side of the equation \( x^2 + 20x \) and check if it equals \( 69 \):

1. For \( x = -23 \):
\[
x^2 + 20x = (-23)^2 + 20(-23) = 529 - 460 = 69 \quad \text{(True)}
\]

2. For \( x = 0 \):
\[
x^2 + 20x = 0^2 + 20(0) = 0 + 0 = 0 \quad \text{(False)}
\]

3. For \( x = 12 \):
\[
x^2 + 20x = 12^2 + 20(12) = 144 + 240 = 384 \quad \text{(False)}
\]

Now, we can summarize the evaluations:

| x | Substituted | Evaluate | True |
|-----|---------------------------------|--------------------------|-------|
| -23 | \((-23)^2 + 20(-23) = 69\) | \(69\) | True |
| 0 | \(0^2 + 20(0) = 0\) | \(0\) | False |
| 12 | \(12^2 + 20(12) = 384\) | \(384\) | False |

So, the filled-in table will look like this:

| x | Substituted | Evaluate | True |
|-----|---------------------------------|--------------------------|-------|
| -23 | \( (-23)^2 + 20(-23) = 69 \) | \( 69 \) | True |
| 0 | \( 0^2 + 20(0) = 0 \) | \( 0 \) | False |
| 12 | \( 12^2 + 20(12) = 384 \) | \( 384 \) | False |