Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![**Solve the Equation**
Given:
\[ x - \sqrt{24 - 5x} = 0 \]
To solve this equation, we need to isolate \( x \) by first dealing with the square root. This involves manipulating the equation to eliminate the square root and then solving the resulting equation for \( x \).
### Steps to Solve:
1. **Isolate the Square Root:**
\[ x = \sqrt{24 - 5x} \]
2. **Square Both Sides:**
\[ x^2 = 24 - 5x \]
3. **Rearrange the Equation:**
\[ x^2 + 5x - 24 = 0 \]
4. **Solve the Quadratic Equation:**
Use the quadratic formula where \( a = 1 \), \( b = 5 \), and \( c = -24 \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
5. **Calculate:**
\[
x = \frac{-5 \pm \sqrt{5^2 - 4 \times 1 \times (-24)}}{2 \times 1}
\]
\[
x = \frac{-5 \pm \sqrt{25 + 96}}{2}
\]
\[
x = \frac{-5 \pm \sqrt{121}}{2}
\]
\[
x = \frac{-5 \pm 11}{2}
\]
6. **Find Solutions:**
- \( x = \frac{-5 + 11}{2} = 3 \)
- \( x = \frac{-5 - 11}{2} = -8 \)
7. **Verify Each Solution:**
Check both solutions in the original equation to ensure they are valid. Remember, squaring can introduce extraneous solutions.
### Note:
Only the solutions that satisfy the original equation without resulting in a negative number under the square root are valid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a204420-1f1f-493e-91ef-fe73ffdaeba1%2Ff0ac1114-e52c-48e7-9965-6c8b2c09ae1f%2Fnct9z0n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solve the Equation**
Given:
\[ x - \sqrt{24 - 5x} = 0 \]
To solve this equation, we need to isolate \( x \) by first dealing with the square root. This involves manipulating the equation to eliminate the square root and then solving the resulting equation for \( x \).
### Steps to Solve:
1. **Isolate the Square Root:**
\[ x = \sqrt{24 - 5x} \]
2. **Square Both Sides:**
\[ x^2 = 24 - 5x \]
3. **Rearrange the Equation:**
\[ x^2 + 5x - 24 = 0 \]
4. **Solve the Quadratic Equation:**
Use the quadratic formula where \( a = 1 \), \( b = 5 \), and \( c = -24 \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
5. **Calculate:**
\[
x = \frac{-5 \pm \sqrt{5^2 - 4 \times 1 \times (-24)}}{2 \times 1}
\]
\[
x = \frac{-5 \pm \sqrt{25 + 96}}{2}
\]
\[
x = \frac{-5 \pm \sqrt{121}}{2}
\]
\[
x = \frac{-5 \pm 11}{2}
\]
6. **Find Solutions:**
- \( x = \frac{-5 + 11}{2} = 3 \)
- \( x = \frac{-5 - 11}{2} = -8 \)
7. **Verify Each Solution:**
Check both solutions in the original equation to ensure they are valid. Remember, squaring can introduce extraneous solutions.
### Note:
Only the solutions that satisfy the original equation without resulting in a negative number under the square root are valid.
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