To solve the equation [tex]\(\sqrt{5x - 1} - 2 = 0\)[/tex], we need to follow a series of algebraic steps. Let’s solve the equation step-by-step:
1. Isolate the square root term:
[tex]\[
\sqrt{5x - 1} - 2 = 0
\][/tex]
Add 2 to both sides to isolate the square root term:
[tex]\[
\sqrt{5x - 1} = 2
\][/tex]
2. Remove the square root:
Square both sides of the equation to eliminate the square root:
[tex]\[
(\sqrt{5x - 1})^2 = 2^2
\][/tex]
Simplifying both sides:
[tex]\[
5x - 1 = 4
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 1 to both sides to isolate the [tex]\(x\)[/tex] term:
[tex]\[
5x = 4 + 1
\][/tex]
Simplifying the right-hand side:
[tex]\[
5x = 5
\][/tex]
Finally, divide both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{5}{5} = 1
\][/tex]
4. Verify the solution:
Substitute [tex]\(x = 1\)[/tex] back into the original equation to verify it satisfies the equation:
[tex]\[
\sqrt{5(1) - 1} - 2 = 0
\][/tex]
Calculate the expression inside the square root:
[tex]\[
\sqrt{5 - 1} - 2 = \sqrt{4} - 2 = 2 - 2 = 0
\][/tex]
Since the left-hand side equals the right-hand side, the solution [tex]\(x = 1\)[/tex] is correct.
Therefore, the solution to the equation [tex]\(\sqrt{5x - 1} - 2 = 0\)[/tex] is
[tex]\[
x = 1
\][/tex]
