Sure, let's find the zeroes of the function [tex]\( f(x) = \sqrt{5x - 1} - 2 \)[/tex].
Step 1: Set the function equal to zero.
To find the zeroes of the function, we set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \sqrt{5x - 1} - 2 = 0 \][/tex]
Step 2: Solve for [tex]\( \sqrt{5x - 1} \)[/tex].
[tex]\[ \sqrt{5x - 1} = 2 \][/tex]
Step 3: Square both sides to eliminate the square root.
[tex]\[ (\sqrt{5x - 1})^2 = 2^2 \][/tex]
[tex]\[ 5x - 1 = 4 \][/tex]
Step 4: Solve for [tex]\( x \)[/tex].
Add 1 to both sides:
[tex]\[ 5x = 5 \][/tex]
Divide both sides by 5:
[tex]\[ x = 1 \][/tex]
Step 5: Verify the solution [tex]\( x = 1 \)[/tex] in the original equation.
[tex]\[ f(1) = \sqrt{5(1) - 1} - 2 \][/tex]
[tex]\[ \quad \ = \sqrt{5 - 1} - 2 \][/tex]
[tex]\[ \quad \ = \sqrt{4} - 2 \][/tex]
[tex]\[ \quad \ = 2 - 2 \][/tex]
[tex]\[ \quad \ = 0 \][/tex]
Since substituting [tex]\( x = 1 \)[/tex] back into the original function gives [tex]\( f(1) = 0 \)[/tex], we confirm that [tex]\( x = 1 \)[/tex] is indeed a zero of the function.
Therefore, the zero of the function [tex]\( f(x) = \sqrt{5x - 1} - 2 \)[/tex] is:
[tex]\[ x = 1 \][/tex]
