Check whether the equation `(x+1)^(3)=x^(3)+x+6` is a quadratic equation or not.

Options

A. yes, it is a quadratic equation
B. it will be cubic equation
C. it will be a linear equation
D. none of these

Answer

To determine whether the equation \((x+1)^{3} = x^{3} + x + 6\) is a quadratic equation, we will start by expanding the left-hand side and simplifying the equation.

### Step 1: Expand the left-hand side
Using the formula for the cube of a binomial, \((a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)\), we can expand \((x + 1)^{3}\):
\[ (x + 1)^{3} = x^{3} + 1^{3} + 3(x)(1)(x + 1) \]
This simplifies to:
\[ x^{3} + 1 + 3x^{2} + 3x \]
So, we have:
\[ (x + 1)^{3} = x^{3} + 3x^{2} + 3x + 1 \]

### Step 2: Set the expanded form equal to the right-hand side
Now, we can set the expanded left-hand side equal to the right-hand side:
\[ x^{3} + 3x^{2} + 3x + 1 = x^{3} + x + 6 \]

### Step 3: Simplify the equation
Next, we will subtract \(x^{3}\) from both sides:
\[ 3x^{2} + 3x + 1 = x + 6 \]
Now, subtract \(x\) and \(6\) from both sides:
\[ 3x^{2} + 3x - x + 1 - 6 = 0 \]
This simplifies to:
\[ 3x^{2} + 2x - 5 = 0 \]

### Step 4: Identify the degree of the polynomial
The equation \(3x^{2} + 2x - 5 = 0\) is now in standard form. The highest power of \(x\) in this equation is \(2\), which indicates that this is a quadratic equation.

### Conclusion
Since the highest degree of the polynomial is \(2\), we conclude that the equation \((x + 1)^{3} = x^{3} + x + 6\) is indeed a quadratic equation.