Question
Solve the following quadratic equation using square roots: (x−7)^2−9/4=0
.(1 point)
Responses
{2,−16}
left brace 2 comma negative 16 right brace
{10,4}
left brace 10 comma 4 right brace
{10.61,3.39}
left brace 10.61 comma 3.39 right brace
{−4,−10}
.(1 point)
Responses
{2,−16}
left brace 2 comma negative 16 right brace
{10,4}
left brace 10 comma 4 right brace
{10.61,3.39}
left brace 10.61 comma 3.39 right brace
{−4,−10}
Answers
Answer
no, thats not what it is Top fraction (x−7)^2−9 / bottom fraction : 4 and last: =0
Answered by
GPT-4o mini
To solve the quadratic equation \((x - 7)^2 - \frac{9}{4} = 0\) using square roots, we can follow these steps:
1. **Isolate the squared term**:
\[
(x - 7)^2 = \frac{9}{4}
\]
2. **Take the square root of both sides**:
\[
x - 7 = \pm \sqrt{\frac{9}{4}}
\]
\[
x - 7 = \pm \frac{3}{2}
\]
3. **Solve for \(x\)**:
- For the positive case:
\[
x - 7 = \frac{3}{2}
\]
\[
x = 7 + \frac{3}{2} = 7 + 1.5 = 8.5
\]
- For the negative case:
\[
x - 7 = -\frac{3}{2}
\]
\[
x = 7 - \frac{3}{2} = 7 - 1.5 = 5.5
\]
Thus, the solutions are \(x = 8.5\) and \(x = 5.5\).
None of the responses match, as it appears there may be a disagreement with the outputs provided.
However, if you are looking for numerical solutions in decimal format:
1. \(8.5\) can be approximated as \(8.5\) or expressed as a fraction which doesn't precisely match any of the options given.
2. \(5.5\) could fit between \(4\) and \(10\), however still, fractions or decimals don't match any given options with exactness.
Make sure to double-check the equation provided or give the correct answers if they were incorrectly formatted in the options you’ve submitted.
1. **Isolate the squared term**:
\[
(x - 7)^2 = \frac{9}{4}
\]
2. **Take the square root of both sides**:
\[
x - 7 = \pm \sqrt{\frac{9}{4}}
\]
\[
x - 7 = \pm \frac{3}{2}
\]
3. **Solve for \(x\)**:
- For the positive case:
\[
x - 7 = \frac{3}{2}
\]
\[
x = 7 + \frac{3}{2} = 7 + 1.5 = 8.5
\]
- For the negative case:
\[
x - 7 = -\frac{3}{2}
\]
\[
x = 7 - \frac{3}{2} = 7 - 1.5 = 5.5
\]
Thus, the solutions are \(x = 8.5\) and \(x = 5.5\).
None of the responses match, as it appears there may be a disagreement with the outputs provided.
However, if you are looking for numerical solutions in decimal format:
1. \(8.5\) can be approximated as \(8.5\) or expressed as a fraction which doesn't precisely match any of the options given.
2. \(5.5\) could fit between \(4\) and \(10\), however still, fractions or decimals don't match any given options with exactness.
Make sure to double-check the equation provided or give the correct answers if they were incorrectly formatted in the options you’ve submitted.
Answered by
GPT-4o mini
Thank you for the clarification! You're working with the equation:
\[
\frac{(x - 7)^2 - 9}{4} = 0
\]
Let's solve this step by step.
1. **Multiply both sides by 4 to eliminate the fraction**:
\[
(x - 7)^2 - 9 = 0
\]
2. **Isolate the squared term**:
\[
(x - 7)^2 = 9
\]
3. **Take the square root of both sides**:
\[
x - 7 = \pm 3
\]
4. **Solve for \(x\)**:
- For the positive case:
\[
x - 7 = 3 \implies x = 10
\]
- For the negative case:
\[
x - 7 = -3 \implies x = 4
\]
Thus, the solutions to the equation are \(x = 10\) and \(x = 4\).
So, the correct answer is:
\(\{10, 4\}\)
This corresponds to the response option:
\(\{10, 4\}\) (left brace 10 comma 4 right brace).
\[
\frac{(x - 7)^2 - 9}{4} = 0
\]
Let's solve this step by step.
1. **Multiply both sides by 4 to eliminate the fraction**:
\[
(x - 7)^2 - 9 = 0
\]
2. **Isolate the squared term**:
\[
(x - 7)^2 = 9
\]
3. **Take the square root of both sides**:
\[
x - 7 = \pm 3
\]
4. **Solve for \(x\)**:
- For the positive case:
\[
x - 7 = 3 \implies x = 10
\]
- For the negative case:
\[
x - 7 = -3 \implies x = 4
\]
Thus, the solutions to the equation are \(x = 10\) and \(x = 4\).
So, the correct answer is:
\(\{10, 4\}\)
This corresponds to the response option:
\(\{10, 4\}\) (left brace 10 comma 4 right brace).