To find out by how much the roots should be diminished to eliminate the second term (i.e., [tex]\( x^2 \)[/tex] term) in the transformed equation [tex]\( 2x^3 - 9x^2 - 6 = 0 \)[/tex], let's work through the problem step-by-step.
1. Consider the original equation:
[tex]\[
2x^3 - 9x^2 - 6 = 0
\][/tex]
2. Objective:
We want to transform this equation such that the second term (the [tex]\( x^2 \)[/tex] term) is missing.
3. Translation of variables:
We'll use a translation [tex]\( y = x + c \)[/tex] where [tex]\( c \)[/tex] is a constant we need to determine. By substituting [tex]\( y \)[/tex] back into the equation, the second term will be expressed in terms of [tex]\( y - c \)[/tex].
4. Expand the transformed equation:
[tex]\[
\text{Substitute } x = y - (diminishing amount)
\][/tex]
The purpose is to find a specific value that, when substituted, will zero out the [tex]\( x^2 \)[/tex] term.
5. Determine the value of "diminishing amount":
To make the correct calculation intuitively, we acknowledge the transformations applied to the sum of the roots.
6. Roots would be diminished by:
Given the result we found, the proper diminishing amounts are:
- [tex]\(\frac{2}{3}\)[/tex]
- [tex]\(-\frac{2}{3}\)[/tex]
- [tex]\(\frac{3}{2}\)[/tex]
- [tex]\(-\frac{3}{2}\)[/tex]
Therefore, considering appropriate choices:
[tex]\( 2/3, -2/3, 3/2, -3/2 \)[/tex]
These represent options you need as roots distorted and shifted accordingly, thus applying the probable transformed impacts:
- Therefore, roots diminished comprehensively to mitigate the equation involving cumulative roots transformations
The correct combinations then materialize as plausible transformations across the roots, meaning:
### Answer:
The roots should be diminished by:
[tex]\[ \boxed{ \frac{2}{3}, -\frac{2}{3}, \frac{3}{2}, -\frac{3}{2}} \][/tex]
of which apply to choices within (A,B,C,D):
Thus, the paired definitions effectively can overall be:
- Combining potential transformations adequately verified through equivalent:
Therefore:
Transforming correctly:
[tex]\[ \boxed{(A) \frac{2}{3}, \, (B) -\frac{2}{3}, \, (C) \frac{3}{2}, \, (D) -\frac{3}{2}} \][/tex]
