To determine which value(s) is/are in the solution set for the quadratic equation \(x^2 + 20x = 69\), we first need to rearrange the equation:
\[ x^2 + 20x - 69 = 0 \]
Next, we will substitute each value (-23, 0, and 12) into the equation and evaluate to see if the equation holds true (i.e., equals 0).
Testing each value
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For \(x = -23\): \[ (-23)^2 + 20(-23) - 69 = 529 - 460 - 69 = 0 \] Thus, the statement is true.
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For \(x = 0\): \[ (0)^2 + 20(0) - 69 = 0 + 0 - 69 = -69 \] Thus, the statement is false.
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For \(x = 12\): \[ (12)^2 + 20(12) - 69 = 144 + 240 - 69 = 315 \] Thus, the statement is false.
Now, we summarize the evaluations in the table:
| \(x\) | Substituted | Evaluate | True Statement? | |---------|---------------------------|--------------------------|------------------| | -23 | \((-23)^2 + 20(-23) - 69\) | \(0\) | True | | 0 | \(0^2 + 20(0) - 69\) | \(-69\) | False | | 12 | \(12^2 + 20(12) - 69\) | \(315\) | False |
Conclusion
Only \(x = -23\) is in the solution set.
So, the final answer is: Only \(x = -23\) is in the solution set.