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Use inspection to solve the equation x^2 = 81, then select the complete solution set below, If there are no real solutions, sel...Question
Use inspection to solve the equation x2=36/169 , then select the correct solution set below. If there are no real solutions, select “There are no real solutions.”(1 point)
Answers
Answer
Solve the following quadratic equation using square roots: (x−18)2=196 .(1 point)
Responses
{32,4}
left brace 32 comma 4 right brace
{214,−178}
left brace 214 comma negative 178 right brace
{21,5}
left brace 21 comma 5 right brace
{−4,−32}
Responses
{32,4}
left brace 32 comma 4 right brace
{214,−178}
left brace 214 comma negative 178 right brace
{21,5}
left brace 21 comma 5 right brace
{−4,−32}
Answer
Solve the following quadratic equation using square roots: (x+6)22=8 .(1 point)
Responses
{−1.17,−6.83}
left brace negative 1.17 comma negative 6.83 right brace
{−2,−10}
left brace negative 2 comma negative 10 right brace
{−0.34,−11.66}
left brace negative 0.34 comma negative 11.66 right brace
{10,−22}
Responses
{−1.17,−6.83}
left brace negative 1.17 comma negative 6.83 right brace
{−2,−10}
left brace negative 2 comma negative 10 right brace
{−0.34,−11.66}
left brace negative 0.34 comma negative 11.66 right brace
{10,−22}
Answer
Solve the following quadratic equation using square roots: (x+36)2−38=251 .(1 point)
Responses
Responses
Answered by
GPT-4o mini
To solve the equation \( x^2 = \frac{36}{169} \) by inspection, we can take the square root of both sides.
The square root of \( x^2 \) is \( x \) or \( -x \), and the square root of \( \frac{36}{169} \) can be found as follows:
\[
\sqrt{\frac{36}{169}} = \frac{\sqrt{36}}{\sqrt{169}} = \frac{6}{13}
\]
Thus, we have:
\[
x = \frac{6}{13} \quad \text{or} \quad x = -\frac{6}{13}
\]
The solution set is:
\[
\left\{ \frac{6}{13}, -\frac{6}{13} \right\}
\]
If these are the possible answers, you should select the option that represents this solution set.
The square root of \( x^2 \) is \( x \) or \( -x \), and the square root of \( \frac{36}{169} \) can be found as follows:
\[
\sqrt{\frac{36}{169}} = \frac{\sqrt{36}}{\sqrt{169}} = \frac{6}{13}
\]
Thus, we have:
\[
x = \frac{6}{13} \quad \text{or} \quad x = -\frac{6}{13}
\]
The solution set is:
\[
\left\{ \frac{6}{13}, -\frac{6}{13} \right\}
\]
If these are the possible answers, you should select the option that represents this solution set.
Answered by
GPT-4o mini
To solve the quadratic equation \((x - 18)^2 = 196\) using square roots, we can follow these steps:
1. Take the square root of both sides of the equation:
\[
x - 18 = \pm \sqrt{196}
\]
2. Since \(\sqrt{196} = 14\), we can rewrite the equation as:
\[
x - 18 = 14 \quad \text{or} \quad x - 18 = -14
\]
3. Solve for \(x\) in each case:
- For \(x - 18 = 14\):
\[
x = 14 + 18 = 32
\]
- For \(x - 18 = -14\):
\[
x = -14 + 18 = 4
\]
Thus, the solutions are \(x = 32\) and \(x = 4\).
The solution set is:
\[
\{32, 4\}
\]
The correct response from your options is:
\[
\{32, 4\}
\]
1. Take the square root of both sides of the equation:
\[
x - 18 = \pm \sqrt{196}
\]
2. Since \(\sqrt{196} = 14\), we can rewrite the equation as:
\[
x - 18 = 14 \quad \text{or} \quad x - 18 = -14
\]
3. Solve for \(x\) in each case:
- For \(x - 18 = 14\):
\[
x = 14 + 18 = 32
\]
- For \(x - 18 = -14\):
\[
x = -14 + 18 = 4
\]
Thus, the solutions are \(x = 32\) and \(x = 4\).
The solution set is:
\[
\{32, 4\}
\]
The correct response from your options is:
\[
\{32, 4\}
\]
Answered by
GPT-4o mini
To solve the equation \((x + 6)^2 = 8\) using square roots, we follow these steps:
1. Take the square root of both sides:
\[
x + 6 = \pm \sqrt{8}
\]
2. Simplifying \(\sqrt{8}\):
\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \approx 2.83
\]
3. Now we can rewrite the equation:
\[
x + 6 = 2\sqrt{2} \quad \text{or} \quad x + 6 = -2\sqrt{2}
\]
4. Solve for \(x\) in each case:
- For \(x + 6 = 2\sqrt{2}\):
\[
x = 2\sqrt{2} - 6 \approx 2.83 - 6 \approx -3.17
\]
- For \(x + 6 = -2\sqrt{2}\):
\[
x = -2\sqrt{2} - 6 \approx -2.83 - 6 \approx -8.83
\]
Thus, the approximate solutions are:
\[
x \approx -3.17 \quad \text{and} \quad x \approx -8.83
\]
Now, looking at the options provided, none of the options represent these exact values, but we can check if we made an error in the approximation. In a more accurate representation:
- \(-3.17\) might be close to \(-1.17\) or \(-0.34\).
- \(-8.83\) is far from \(-6.83\) or \(-10\).
Let's convert these approximate values to decimal for clarity:
The exact roots based on the radicals would give a precise calculation:
- \(-6 + 2\sqrt{2} \approx -3.17\)
- \(-6 - 2\sqrt{2} \approx -8.83\)
Now observing the options:
1. \(\{-1.17, -6.83\}\)
2. \(\{-2, -10\}\)
3. \(\{-0.34, -11.66\}\)
4. \(\{10, -22\}\)
The closest set based on our approximation is none of these perfectly match \(-3.17\) and \(-8.83\), yet \(\{-2, -10\}\) may represent approximate values.
Since the question may not directly yield those specific results, a clarification of the approximations might lean to consider none are satisfactory directly.
If you had to choose the best option, reconsider your equation setup or confirm choice relevance.
Double-check your computations against unique decimal forms and supplied roots accordingly.
1. Take the square root of both sides:
\[
x + 6 = \pm \sqrt{8}
\]
2. Simplifying \(\sqrt{8}\):
\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \approx 2.83
\]
3. Now we can rewrite the equation:
\[
x + 6 = 2\sqrt{2} \quad \text{or} \quad x + 6 = -2\sqrt{2}
\]
4. Solve for \(x\) in each case:
- For \(x + 6 = 2\sqrt{2}\):
\[
x = 2\sqrt{2} - 6 \approx 2.83 - 6 \approx -3.17
\]
- For \(x + 6 = -2\sqrt{2}\):
\[
x = -2\sqrt{2} - 6 \approx -2.83 - 6 \approx -8.83
\]
Thus, the approximate solutions are:
\[
x \approx -3.17 \quad \text{and} \quad x \approx -8.83
\]
Now, looking at the options provided, none of the options represent these exact values, but we can check if we made an error in the approximation. In a more accurate representation:
- \(-3.17\) might be close to \(-1.17\) or \(-0.34\).
- \(-8.83\) is far from \(-6.83\) or \(-10\).
Let's convert these approximate values to decimal for clarity:
The exact roots based on the radicals would give a precise calculation:
- \(-6 + 2\sqrt{2} \approx -3.17\)
- \(-6 - 2\sqrt{2} \approx -8.83\)
Now observing the options:
1. \(\{-1.17, -6.83\}\)
2. \(\{-2, -10\}\)
3. \(\{-0.34, -11.66\}\)
4. \(\{10, -22\}\)
The closest set based on our approximation is none of these perfectly match \(-3.17\) and \(-8.83\), yet \(\{-2, -10\}\) may represent approximate values.
Since the question may not directly yield those specific results, a clarification of the approximations might lean to consider none are satisfactory directly.
If you had to choose the best option, reconsider your equation setup or confirm choice relevance.
Double-check your computations against unique decimal forms and supplied roots accordingly.
Answered by
GPT-4o mini
Let's solve the equation \((x + 36)^2 - 38 = 251\).
1. Start by isolating the squared term:
\[
(x + 36)^2 - 38 + 38 = 251 + 38
\]
This simplifies to:
\[
(x + 36)^2 = 289
\]
2. Take the square root of both sides:
\[
x + 36 = \pm \sqrt{289}
\]
Since \(\sqrt{289} = 17\), we can write:
\[
x + 36 = 17 \quad \text{or} \quad x + 36 = -17
\]
3. Now solve for \(x\) in both cases:
- For \(x + 36 = 17\):
\[
x = 17 - 36 = -19
\]
- For \(x + 36 = -17\):
\[
x = -17 - 36 = -53
\]
Thus, the solutions to the equation are:
\[
x = -19 \quad \text{and} \quad x = -53
\]
So the solution set is:
\[
\{-19, -53\}
\]
If there are specific response options provided, please share them, and I can confirm if this solution aligns with one of your choices.
1. Start by isolating the squared term:
\[
(x + 36)^2 - 38 + 38 = 251 + 38
\]
This simplifies to:
\[
(x + 36)^2 = 289
\]
2. Take the square root of both sides:
\[
x + 36 = \pm \sqrt{289}
\]
Since \(\sqrt{289} = 17\), we can write:
\[
x + 36 = 17 \quad \text{or} \quad x + 36 = -17
\]
3. Now solve for \(x\) in both cases:
- For \(x + 36 = 17\):
\[
x = 17 - 36 = -19
\]
- For \(x + 36 = -17\):
\[
x = -17 - 36 = -53
\]
Thus, the solutions to the equation are:
\[
x = -19 \quad \text{and} \quad x = -53
\]
So the solution set is:
\[
\{-19, -53\}
\]
If there are specific response options provided, please share them, and I can confirm if this solution aligns with one of your choices.
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