The transformed equation of `x^3 -6x^2 +5x +8=0` by eliminating second term is

Options

A. `x^3 +7x +2=0`
B. `x^3 -7x +2=0`
C. `x^3 + 5x +2=0`
D. `x^3 - 5x +2=0`

Answer

To transform the equation \( x^3 - 6x^2 + 5x + 8 = 0 \) by eliminating the second term, we can follow these steps:

### Step 1: Write the given equation
The original equation is:
\[ x^3 - 6x^2 + 5x + 8 = 0 \]

### Step 2: Substitute \( x = x + h \)
We substitute \( x \) with \( x + h \) to eliminate the \( x^2 \) term:
\[ (x + h)^3 - 6(x + h)^2 + 5(x + h) + 8 = 0 \]

### Step 3: Expand the equation
Now we expand the terms:
\[ (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \]
\[ -6(x + h)^2 = -6(x^2 + 2xh + h^2) = -6x^2 - 12xh - 6h^2 \]
\[ 5(x + h) = 5x + 5h \]
Putting it all together:
\[ x^3 + 3x^2h + 3xh^2 + h^3 - 6x^2 - 12xh - 6h^2 + 5x + 5h + 8 = 0 \]

### Step 4: Combine like terms
Combining the terms gives:
\[ x^3 + (3h - 6)x^2 + (3h^2 - 12h + 5)x + (h^3 - 6h^2 + 5h + 8) = 0 \]

### Step 5: Set the coefficient of \( x^2 \) to zero
To eliminate the \( x^2 \) term, we set the coefficient \( 3h - 6 = 0 \):
\[ 3h - 6 = 0 \implies h = 2 \]

### Step 6: Substitute \( h \) back into the equation
Now substitute \( h = 2 \) back into the equation:
\[ x^3 + (3(2)^2 - 12(2) + 5)x + (2^3 - 6(2^2) + 5(2) + 8) = 0 \]
Calculating the coefficients:
- For \( x \):
\[ 3(2^2) - 12(2) + 5 = 12 - 24 + 5 = -7 \]
- For the constant term:
\[ 2^3 - 6(2^2) + 5(2) + 8 = 8 - 24 + 10 + 8 = 2 \]

### Step 7: Write the transformed equation
Thus, the transformed equation is:
\[ x^3 - 7x + 2 = 0 \]

### Final Answer
The transformed equation of \( x^3 - 6x^2 + 5x + 8 = 0 \) by eliminating the second term is:
\[ \boxed{x^3 - 7x + 2 = 0} \]