Chek whether the following are quadratic equations :
`x^(3) - 4x^(2) - x + 1 = (x- 2)^(3)`

Answer

To check whether the given polynomial \( x^3 - 4x^2 - x + 1 = (x - 2)^3 \) is a quadratic equation, we will simplify the equation step by step.

### Step 1: Expand the right-hand side
We start by expanding \( (x - 2)^3 \) using the formula for the cube of a binomial:
\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]
Here, \( a = x \) and \( b = 2 \):
\[ (x - 2)^3 = x^3 - 3x^2 \cdot 2 + 3x \cdot 2^2 - 2^3 \]
Calculating this gives:
\[ = x^3 - 6x^2 + 12x - 8 \]

### Step 2: Set the equation
Now we can set the left-hand side equal to the expanded right-hand side:
\[ x^3 - 4x^2 - x + 1 = x^3 - 6x^2 + 12x - 8 \]

### Step 3: Move all terms to one side
Subtract \( x^3 \) from both sides:
\[ -4x^2 - x + 1 = -6x^2 + 12x - 8 \]
Now, add \( 6x^2 \) and subtract \( 12x \) and add \( 8 \) to both sides:
\[ -4x^2 + 6x^2 - x - 12x + 1 + 8 = 0 \]
This simplifies to:
\[ 2x^2 - 13x + 9 = 0 \]

### Step 4: Identify the degree of the polynomial
The highest power of \( x \) in the equation \( 2x^2 - 13x + 9 = 0 \) is 2.

### Conclusion
Since the highest degree of the polynomial is 2, we conclude that the given polynomial is indeed a quadratic equation.

### Final Answer
Yes, \( x^3 - 4x^2 - x + 1 = (x - 2)^3 \) simplifies to a quadratic equation \( 2x^2 - 13x + 9 = 0 \).

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