Question
Use the table to answer the question. x Substituted Evaluate True Statement? −23 0 12 For the quadratic equation x2+20x=69 , complete the table by testing each of the given values to determine whether it is a solution. Identify which one of these values is in the solution set. (1 point) Only x=blank is in the solution set.
Answers
GPT-4o mini
To determine which value is in the solution set of the quadratic equation \(x^2 + 20x = 69\), we first rearrange the equation to standard form:
\[
x^2 + 20x - 69 = 0
\]
Next, we will evaluate this equation for the given values: \(x = -23\), \(x = 0\), and \(x = 12\).
1. **For \(x = -23\)**:
\[
(-23)^2 + 20(-23) - 69 = 529 - 460 - 69 = 0
\]
This is a true statement.
2. **For \(x = 0\)**:
\[
(0)^2 + 20(0) - 69 = 0 - 69 = -69
\]
This is not a true statement.
3. **For \(x = 12\)**:
\[
(12)^2 + 20(12) - 69 = 144 + 240 - 69 = 315
\]
This is not a true statement.
After evaluating all three values, we find only \(x = -23\) results in the equation being equal to zero.
Thus, the answer is:
Only \(x = -23\) is in the solution set.
\[
x^2 + 20x - 69 = 0
\]
Next, we will evaluate this equation for the given values: \(x = -23\), \(x = 0\), and \(x = 12\).
1. **For \(x = -23\)**:
\[
(-23)^2 + 20(-23) - 69 = 529 - 460 - 69 = 0
\]
This is a true statement.
2. **For \(x = 0\)**:
\[
(0)^2 + 20(0) - 69 = 0 - 69 = -69
\]
This is not a true statement.
3. **For \(x = 12\)**:
\[
(12)^2 + 20(12) - 69 = 144 + 240 - 69 = 315
\]
This is not a true statement.
After evaluating all three values, we find only \(x = -23\) results in the equation being equal to zero.
Thus, the answer is:
Only \(x = -23\) is in the solution set.
Honeybeee
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=blank. The larger of the two solutions is x=blank.
GPT-4o mini
To solve the quadratic equation \((x - 7)^2 = 11\), we first take the square root of both sides:
1. Rearranging the equation gives:
\[
x - 7 = \pm \sqrt{11}
\]
2. This results in two equations:
\[
x - 7 = \sqrt{11} \quad \text{and} \quad x - 7 = -\sqrt{11}
\]
3. Solving each equation for \(x\):
- **First equation**:
\[
x = 7 + \sqrt{11}
\]
- **Second equation**:
\[
x = 7 - \sqrt{11}
\]
Now we have the two potential solutions:
- The larger solution is \(x = 7 + \sqrt{11}\).
- The smaller solution is \(x = 7 - \sqrt{11}\).
Next, we can evaluate these solutions using approximate decimal values:
- The approximate value of \(\sqrt{11} \approx 3.3166\).
Therefore:
- **Larger solution**:
\[
x = 7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166
\]
- **Smaller solution**:
\[
x = 7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834
\]
Finally, we can fill in the blanks:
The smaller of the two solutions is \(x = 7 - \sqrt{11}\) (approximately \(3.6834\)), and the larger of the two solutions is \(x = 7 + \sqrt{11}\) (approximately \(10.3166\)).
Thus, the final answers are:
The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
The larger of the two solutions is \(x = 7 + \sqrt{11}\).
1. Rearranging the equation gives:
\[
x - 7 = \pm \sqrt{11}
\]
2. This results in two equations:
\[
x - 7 = \sqrt{11} \quad \text{and} \quad x - 7 = -\sqrt{11}
\]
3. Solving each equation for \(x\):
- **First equation**:
\[
x = 7 + \sqrt{11}
\]
- **Second equation**:
\[
x = 7 - \sqrt{11}
\]
Now we have the two potential solutions:
- The larger solution is \(x = 7 + \sqrt{11}\).
- The smaller solution is \(x = 7 - \sqrt{11}\).
Next, we can evaluate these solutions using approximate decimal values:
- The approximate value of \(\sqrt{11} \approx 3.3166\).
Therefore:
- **Larger solution**:
\[
x = 7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166
\]
- **Smaller solution**:
\[
x = 7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834
\]
Finally, we can fill in the blanks:
The smaller of the two solutions is \(x = 7 - \sqrt{11}\) (approximately \(3.6834\)), and the larger of the two solutions is \(x = 7 + \sqrt{11}\) (approximately \(10.3166\)).
Thus, the final answers are:
The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
The larger of the two solutions is \(x = 7 + \sqrt{11}\).
Honeybeee
How many solutions would the equation x2=−9 have? You do not need to solve for x.(1 point) The equation would have blank solution(s).
GPT-4o mini
The equation \(x^2 = -9\) cannot have real solutions because the square of a real number cannot be negative. However, if we consider complex numbers, we can find solutions.
For the equation \(x^2 = -9\), we can express it as:
\[
x = \pm \sqrt{-9} = \pm 3i
\]
Thus, in the context of complex numbers, there are 2 solutions: \(3i\) and \(-3i\).
Therefore, the statement would be:
The equation would have **2 solutions**.
For the equation \(x^2 = -9\), we can express it as:
\[
x = \pm \sqrt{-9} = \pm 3i
\]
Thus, in the context of complex numbers, there are 2 solutions: \(3i\) and \(-3i\).
Therefore, the statement would be:
The equation would have **2 solutions**.