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

Answer

To determine whether the equation \((x + 2)^{3} = 2x (x^{2} - 1)\) is a quadratic equation, we will simplify both sides and analyze the resulting equation.

### Step 1: Expand the left-hand side
We start with the left-hand side of the equation:
\[ (x + 2)^{3} \]
Using the binomial expansion formula, we can expand this as follows:
\[ (x + 2)^{3} = x^{3} + 3(x^{2})(2) + 3(x)(2^{2}) + 2^{3} \]
Calculating each term:
- \(x^{3}\)
- \(3 \cdot 2 \cdot x^{2} = 6x^{2}\)
- \(3 \cdot x \cdot 4 = 12x\)
- \(2^{3} = 8\)

So, we have:
\[ (x + 2)^{3} = x^{3} + 6x^{2} + 12x + 8 \]

### Step 2: Expand the right-hand side
Now, we expand the right-hand side:
\[ 2x(x^{2} - 1) = 2x^{3} - 2x \]

### Step 3: Set the equation to zero
Now we set the left-hand side equal to the right-hand side:
\[ x^{3} + 6x^{2} + 12x + 8 = 2x^{3} - 2x \]
Rearranging gives:
\[ x^{3} - 2x^{3} + 6x^{2} + 12x + 2x + 8 = 0 \]
This simplifies to:
\[ -x^{3} + 6x^{2} + 14x + 8 = 0 \]
Or, multiplying through by -1:
\[ x^{3} - 6x^{2} - 14x - 8 = 0 \]

### Step 4: Analyze the degree of the polynomial
The resulting equation is:
\[ x^{3} - 6x^{2} - 14x - 8 = 0 \]
The highest power of \(x\) in this equation is 3, which indicates that this is a cubic equation.

### Conclusion
Since the highest degree of the polynomial is 3, the given equation is **not** a quadratic equation.

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