Asked by Smartypants
Use the table to answer the question.x Substituted Evaluate True Statement?7−11−−√711−7–√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=.
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \((x - 7)^2 = 11\), we first take the square root of both sides:
\[
x - 7 = \pm \sqrt{11}
\]
This gives us two equations to solve:
1. \(x - 7 = \sqrt{11}\)
2. \(x - 7 = -\sqrt{11}\)
Now, we can solve each equation for \(x\):
1. From \(x - 7 = \sqrt{11}\):
\[
x = 7 + \sqrt{11}
\]
2. From \(x - 7 = -\sqrt{11}\):
\[
x = 7 - \sqrt{11}
\]
Now we have our two solutions:
- The smaller solution is \(x = 7 - \sqrt{11}\).
- The larger solution is \(x = 7 + \sqrt{11}\).
### Now let's identify the values explicitly
Given that \(\sqrt{11} \approx 3.32\) (approximately):
1. Smaller solution:
\[
x = 7 - \sqrt{11} \approx 7 - 3.32 = 3.68
\]
2. Larger solution:
\[
x = 7 + \sqrt{11} \approx 7 + 3.32 = 10.32
\]
Thus, the final answers for the two solutions to the quadratic equation are:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).
\[
x - 7 = \pm \sqrt{11}
\]
This gives us two equations to solve:
1. \(x - 7 = \sqrt{11}\)
2. \(x - 7 = -\sqrt{11}\)
Now, we can solve each equation for \(x\):
1. From \(x - 7 = \sqrt{11}\):
\[
x = 7 + \sqrt{11}
\]
2. From \(x - 7 = -\sqrt{11}\):
\[
x = 7 - \sqrt{11}
\]
Now we have our two solutions:
- The smaller solution is \(x = 7 - \sqrt{11}\).
- The larger solution is \(x = 7 + \sqrt{11}\).
### Now let's identify the values explicitly
Given that \(\sqrt{11} \approx 3.32\) (approximately):
1. Smaller solution:
\[
x = 7 - \sqrt{11} \approx 7 - 3.32 = 3.68
\]
2. Larger solution:
\[
x = 7 + \sqrt{11} \approx 7 + 3.32 = 10.32
\]
Thus, the final answers for the two solutions to the quadratic equation are:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).
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