Answer :

GinnyAnswer
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].

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