Remove second term ( second higher power of x ) from the equation
`x^3 -6x^2+10x -3=0`

Answer

To remove the second term (the second higher power of \(x\)) from the equation \(x^3 - 6x^2 + 10x - 3 = 0\), we can use a substitution method. Here’s a step-by-step solution:

### Step 1: Write the original equation
Start with the original equation:
\[ x^3 - 6x^2 + 10x - 3 = 0 \]

### Step 2: Substitute \(x\) with \(y + a\)
To eliminate the \(x^2\) term, we will make the substitution \(x = y + a\), where \(a\) is a constant we will determine later. Substitute this into the equation:
\[ (y + a)^3 - 6(y + a)^2 + 10(y + a) - 3 = 0 \]

### Step 3: Expand the equation
Now, expand the equation:
\[ (y^3 + 3ay^2 + 3a^2y + a^3) - 6(y^2 + 2ay + a^2) + 10y + 10a - 3 = 0 \]
This simplifies to:
\[ y^3 + 3ay^2 + 3a^2y + a^3 - 6y^2 - 12ay - 6a^2 + 10y + 10a - 3 = 0 \]

### Step 4: Collect like terms
Now, group the terms by their powers of \(y\):
\[ y^3 + (3a - 6)y^2 + (3a^2 - 12a + 10)y + (a^3 - 6a^2 + 10a - 3) = 0 \]

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

### Step 6: Substitute \(a\) back into the equation
Now substitute \(a = 2\) back into the equation:
\[ y^3 + (3(2)^2 - 12(2) + 10)y + (2^3 - 6(2^2) + 10(2) - 3) = 0 \]
Calculating the coefficients:
- For \(y\):
\[ 3(4) - 24 + 10 = 12 - 24 + 10 = -2 \]
- For the constant term:
\[ 8 - 24 + 20 - 3 = 1 \]

### Step 7: Write the final equation
Thus, the equation simplifies to:
\[ y^3 - 2y + 1 = 0 \]

This equation does not contain the second power of \(y\), which means we have successfully removed the second term.

### Final Answer:
The transformed equation is:
\[ y^3 - 2y + 1 = 0 \]

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