To transform the given cubic equation [tex]\(2x^3 - 9x^2 - 6 = 0\)[/tex] into a form where the second term is eliminated, we need to apply a specific substitution to [tex]\(x\)[/tex].
### Step-by-Step Procedure
1. Identify the General Form:
Consider the cubic polynomial in the form [tex]\(ax^3 + bx^2 + cx + d = 0\)[/tex]. In our problem, [tex]\(a = 2\)[/tex], [tex]\(b = -9\)[/tex], [tex]\(c = 0\)[/tex], and [tex]\(d = -6\)[/tex].
2. Substitution Method:
To eliminate the second term, [tex]\(bx^2\)[/tex], we use the substitution [tex]\(x = y + h\)[/tex] where [tex]\(h\)[/tex] is a constant that needs to be determined.
3. Perform the Substitution:
Substitute [tex]\(x = y + h\)[/tex] into the equation:
[tex]\[
2(y+h)^3 - 9(y+h)^2 - 6 = 0
\][/tex]
4. Expand the Terms:
Let's expand [tex]\((y+h)^3\)[/tex] and [tex]\((y+h)^2\)[/tex]:
[tex]\[
(y+h)^3 = y^3 + 3hy^2 + 3h^2y + h^3
\][/tex]
[tex]\[
(y+h)^2 = y^2 + 2hy + h^2
\][/tex]
Therefore,
[tex]\[
2(y+h)^3 = 2(y^3 + 3hy^2 + 3h^2y + h^3) = 2y^3 + 6hy^2 + 6h^2y + 2h^3
\][/tex]
[tex]\[
-9(y+h)^2 = -9(y^2 + 2hy + h^2) = -9y^2 - 18hy - 9h^2
\][/tex]
5. Combine the Terms:
Substitute these back into the original equation:
[tex]\[
2(y^3 + 6hy^2 + 6h^2y + 2h^3) - 9(y^2 + 2hy + h^2) - 6 = 0
\][/tex]
Simplify the expression:
[tex]\[
2y^3 + 6hy^2 + 6h^2y + 2h^3 - 9y^2 - 18hy - 9h^2 - 6 = 0
\][/tex]
Group the terms:
[tex]\[
2y^3 + (6h - 9)y^2 + (6h^2 - 18h)y + (2h^3 - 9h^2 - 6) = 0
\][/tex]
6. Eliminate the Second Term:
For the second term to be zero, the coefficient of [tex]\(y^2\)[/tex] must be zero:
[tex]\[
6h - 9 = 0
\][/tex]
Solve for [tex]\(h\)[/tex]:
[tex]\[
6h = 9 \\
h = \frac{9}{6} = \frac{3}{2}
\][/tex]
### Conclusion
The roots are to be diminished by [tex]\(\frac{3}{2}\)[/tex].
Thus, the correct choice is:
(c) [tex]\(\frac{3}{2}\)[/tex]
