If x is a rational number and `((x + 1) ^(3) - ( x - 1) ^(3))/( ( x + 1)^(2) - ( x - 1) ^(2)) = 2 ` then the sum of numerator and denominator of x is

Options

A. 3
B. 4
C. 5
D. 7

Answer

To solve the equation

\[ \frac{(x + 1)^3 - (x - 1)^3}{(x + 1)^2 - (x - 1)^2} = 2, \]

we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify the Numerator
The numerator is \((x + 1)^3 - (x - 1)^3\). We can use the difference of cubes formula, which states that \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Let \(a = x + 1\) and \(b = x - 1\). Then, we have:

\[ a - b = (x + 1) - (x - 1) = 2. \]

Now we need to calculate \(a^2 + ab + b^2\):

\[ a^2 = (x + 1)^2 = x^2 + 2x + 1, \]
\[ b^2 = (x - 1)^2 = x^2 - 2x + 1, \]
\[ ab = (x + 1)(x - 1) = x^2 - 1. \]

Now, adding these together:

\[ a^2 + ab + b^2 = (x^2 + 2x + 1) + (x^2 - 1) + (x^2 - 2x + 1) = 3x^2 + 1. \]

So, the numerator simplifies to:

\[ (x + 1)^3 - (x - 1)^3 = 2(3x^2 + 1) = 6x^2 + 2. \]

### Step 2: Simplify the Denominator
The denominator is \((x + 1)^2 - (x - 1)^2\). We can use the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\).

Let \(a = (x + 1)\) and \(b = (x - 1)\):

\[ a - b = (x + 1) - (x - 1) = 2, \]
\[ a + b = (x + 1) + (x - 1) = 2x. \]

Thus, the denominator simplifies to:

\[ (x + 1)^2 - (x - 1)^2 = 2(2x) = 4x. \]

### Step 3: Substitute Back into the Equation
Now we can substitute the simplified numerator and denominator back into the equation:

\[ \frac{6x^2 + 2}{4x} = 2. \]

### Step 4: Cross Multiply
Cross multiplying gives us:

\[ 6x^2 + 2 = 8x. \]

### Step 5: Rearranging the Equation
Rearranging this equation gives us:

\[ 6x^2 - 8x + 2 = 0. \]

### Step 6: Simplifying the Quadratic Equation
We can simplify this equation by dividing everything by 2:

\[ 3x^2 - 4x + 1 = 0. \]

### Step 7: Solving the Quadratic Equation
Now we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here, \(a = 3\), \(b = -4\), and \(c = 1\):

\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} = \frac{4 \pm \sqrt{16 - 12}}{6} = \frac{4 \pm \sqrt{4}}{6} = \frac{4 \pm 2}{6}. \]

This gives us two solutions:

1. \(x = \frac{6}{6} = 1\)
2. \(x = \frac{2}{6} = \frac{1}{3}\)

### Step 8: Finding the Sum of the Numerator and Denominator of \(x\)
Now we need to find the sum of the numerator and denominator of \(x\) for each solution:

For \(x = 1\):
- Numerator = 1, Denominator = 1 → Sum = \(1 + 1 = 2\).

For \(x = \frac{1}{3}\):
- Numerator = 1, Denominator = 3 → Sum = \(1 + 3 = 4\).

Thus, the possible sums are 2 and 4.

### Final Answer
The sum of the numerator and denominator of \(x\) can be either 2 or 4.

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