Certainly! To find the value of [tex]\( k \)[/tex] given that one root of the quadratic equation [tex]\( 6x^2 - x - k = 0 \)[/tex] is [tex]\( \frac{2}{3} \)[/tex], follow these steps:
1. Substitute the root into the quadratic equation:
Given that [tex]\( x = \frac{2}{3} \)[/tex] is a root, the equation [tex]\( 6x^2 - x - k = 0 \)[/tex] should be satisfied when [tex]\( x = \frac{2}{3} \)[/tex].
2. Substitute [tex]\( x = \frac{2}{3} \)[/tex] into the equation:
[tex]\[
6 \left( \frac{2}{3} \right)^2 - \left( \frac{2}{3} \right) - k = 0
\][/tex]
3. Simplify the equation:
First, calculate [tex]\( \left( \frac{2}{3} \right)^2 \)[/tex]:
[tex]\[
\left( \frac{2}{3} \right)^2 = \frac{4}{9}
\][/tex]
Now, multiply it by 6:
[tex]\[
6 \cdot \frac{4}{9} = \frac{24}{9} = \frac{8}{3}
\][/tex]
Then, substitute all the values into the equation:
[tex]\[
\frac{8}{3} - \frac{2}{3} - k = 0
\][/tex]
4. Combine like terms:
[tex]\[
\frac{8}{3} - \frac{2}{3} = \frac{6}{3} = 2
\][/tex]
So the equation simplifies to:
[tex]\[
2 - k = 0
\][/tex]
5. Solve for [tex]\( k \)[/tex]:
[tex]\[
2 - k = 0 \implies k = 2
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{2} \)[/tex].
