Answer :
To solve the equation [tex]\(c^2 = \frac{144}{169}\)[/tex], we need to find the values of [tex]\(c\)[/tex] where squaring them gives [tex]\(\frac{144}{169}\)[/tex].
Step-by-Step Solution:
1. Understand the Equation:
[tex]\[
c^2 = \frac{144}{169}
\][/tex]
2. Take the Square Root on Both Sides:
To solve for [tex]\(c\)[/tex], we take the square root of both sides of the equation. Remember, taking the square root can result in both positive and negative solutions:
[tex]\[
c = \sqrt{\frac{144}{169}} \quad \text{or} \quad c = -\sqrt{\frac{144}{169}}
\][/tex]
3. Compute the Square Root:
- Calculate the positive square root:
[tex]\[
c = \frac{\sqrt{144}}{\sqrt{169}} = \frac{12}{13}
\][/tex]
- Calculate the negative square root:
[tex]\[
c = -\frac{\sqrt{144}}{\sqrt{169}} = -\frac{12}{13}
\][/tex]
4. Identify the Solutions:
The solutions to the equation [tex]\(c^2 = \frac{144}{169}\)[/tex] are:
[tex]\[
c = \frac{12}{13} \quad \text{and} \quad c = -\frac{12}{13}
\][/tex]
5. Choose the Correct Answers:
From the given options:
- (A) [tex]\(c = \frac{12}{13}\)[/tex] is correct.
- (B) [tex]\(c = -\frac{12}{13}\)[/tex] is also correct.
- (C) [tex]\(c = \frac{14}{15}\)[/tex] and (D) [tex]\(c = -\frac{14}{15}\)[/tex] are incorrect.
- Option (E) None of the above is also incorrect as we found valid solutions.
Therefore, the correct answers are (A) and (B): [tex]\(c = \frac{12}{13}\)[/tex] and [tex]\(c = -\frac{12}{13}\)[/tex].
Step-by-Step Solution:
1. Understand the Equation:
[tex]\[
c^2 = \frac{144}{169}
\][/tex]
2. Take the Square Root on Both Sides:
To solve for [tex]\(c\)[/tex], we take the square root of both sides of the equation. Remember, taking the square root can result in both positive and negative solutions:
[tex]\[
c = \sqrt{\frac{144}{169}} \quad \text{or} \quad c = -\sqrt{\frac{144}{169}}
\][/tex]
3. Compute the Square Root:
- Calculate the positive square root:
[tex]\[
c = \frac{\sqrt{144}}{\sqrt{169}} = \frac{12}{13}
\][/tex]
- Calculate the negative square root:
[tex]\[
c = -\frac{\sqrt{144}}{\sqrt{169}} = -\frac{12}{13}
\][/tex]
4. Identify the Solutions:
The solutions to the equation [tex]\(c^2 = \frac{144}{169}\)[/tex] are:
[tex]\[
c = \frac{12}{13} \quad \text{and} \quad c = -\frac{12}{13}
\][/tex]
5. Choose the Correct Answers:
From the given options:
- (A) [tex]\(c = \frac{12}{13}\)[/tex] is correct.
- (B) [tex]\(c = -\frac{12}{13}\)[/tex] is also correct.
- (C) [tex]\(c = \frac{14}{15}\)[/tex] and (D) [tex]\(c = -\frac{14}{15}\)[/tex] are incorrect.
- Option (E) None of the above is also incorrect as we found valid solutions.
Therefore, the correct answers are (A) and (B): [tex]\(c = \frac{12}{13}\)[/tex] and [tex]\(c = -\frac{12}{13}\)[/tex].
