To solve for [tex]\( k \)[/tex] in the equation [tex]\( 3x(x - 2k) = 3x^2 - 48x \)[/tex], follow these steps:
1. Expand the left-hand side:
[tex]\[
3x(x - 2k) = 3x^2 - 6kx
\][/tex]
2. Compare the expanded left-hand side to the right-hand side:
[tex]\[
3x^2 - 6kx = 3x^2 - 48x
\][/tex]
3. Since both expressions must be equal for all [tex]\( x \)[/tex], compare the coefficients of [tex]\( x \)[/tex] on both sides of the equation:
- Coefficient of [tex]\( x^2 \)[/tex]: [tex]\( 3 \)[/tex]
- Coefficient of [tex]\( x \)[/tex]: [tex]\( -6k \)[/tex]
Therefore, by comparing the coefficients of [tex]\( x \)[/tex]:
[tex]\[
-6k = -48
\][/tex]
4. Solve for [tex]\( k \)[/tex] by isolating [tex]\( k \)[/tex] on one side of the equation:
[tex]\[
-6k = -48
\][/tex]
[tex]\[
k = \frac{-48}{-6}
\][/tex]
[tex]\[
k = 8
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{8} \)[/tex].
