Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) √5x-2-3-0 X =
Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) √5x-2-3-0 X =
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,...
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![**Solving Radical Equations**
**Instruction:**
Solve the radical equation below. Be sure to check all solutions to eliminate extraneous solutions. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
\[ \sqrt{5x - 2} - 3 = 0 \]
**Solution:**
\[ x = \boxed{} \]
*Explanation:*
In this problem, we are given a radical equation featuring a square root. To solve for x, ensure you isolate the radical, square both sides of the equation to eliminate the square root, and then solve for x. Once solutions are found, check each solution in the original equation to verify it is not extraneous.
**Step-by-Step Guide:**
1. **Isolate the Radical:**
Bring the term with the square root on one side of the equation:
\[ \sqrt{5x - 2} = 3 \]
2. **Square Both Sides:**
Squaring both sides to eliminate the square root:
\[ (\sqrt{5x - 2})^2 = 3^2 \]
\[ 5x - 2 = 9 \]
3. **Solve for x:**
Solve the resulting linear equation:
\[ 5x = 11 \]
\[ x = \frac{11}{5} \]
4. **Verify the Solution:**
Substitute \( x = \frac{11}{5} \) back into the original equation to check for extraneous solutions.
\[ \sqrt{5 \left(\frac{11}{5}\right) - 2} - 3 = 0 \]
Simplifying inside the square root:
\[ \sqrt{11 - 2} - 3 = 0 \]
\[ \sqrt{9} - 3 = 0 \]
\[ 3 - 3 = 0 \]
\[ 0 = 0 \]
The solution \( x = \frac{11}{5} \) satisfies the original equation, therefore the solution to the given problem is:
\[ x = \frac{11}{5} \]
In case no solution existed, you would enter DNE (Does Not Exist).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc630d89f-ffd5-405f-84e2-a10f6a309757%2Feb93673c-e89b-4653-98c6-0910491b552e%2Ffbfc4ie_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Radical Equations**
**Instruction:**
Solve the radical equation below. Be sure to check all solutions to eliminate extraneous solutions. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
\[ \sqrt{5x - 2} - 3 = 0 \]
**Solution:**
\[ x = \boxed{} \]
*Explanation:*
In this problem, we are given a radical equation featuring a square root. To solve for x, ensure you isolate the radical, square both sides of the equation to eliminate the square root, and then solve for x. Once solutions are found, check each solution in the original equation to verify it is not extraneous.
**Step-by-Step Guide:**
1. **Isolate the Radical:**
Bring the term with the square root on one side of the equation:
\[ \sqrt{5x - 2} = 3 \]
2. **Square Both Sides:**
Squaring both sides to eliminate the square root:
\[ (\sqrt{5x - 2})^2 = 3^2 \]
\[ 5x - 2 = 9 \]
3. **Solve for x:**
Solve the resulting linear equation:
\[ 5x = 11 \]
\[ x = \frac{11}{5} \]
4. **Verify the Solution:**
Substitute \( x = \frac{11}{5} \) back into the original equation to check for extraneous solutions.
\[ \sqrt{5 \left(\frac{11}{5}\right) - 2} - 3 = 0 \]
Simplifying inside the square root:
\[ \sqrt{11 - 2} - 3 = 0 \]
\[ \sqrt{9} - 3 = 0 \]
\[ 3 - 3 = 0 \]
\[ 0 = 0 \]
The solution \( x = \frac{11}{5} \) satisfies the original equation, therefore the solution to the given problem is:
\[ x = \frac{11}{5} \]
In case no solution existed, you would enter DNE (Does Not Exist).
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