If `x+(a)/(x) =1` then the value of `(x^(3) -x^(2))/(x^(2) +x +a)` in terms of a is

Options

A. `(a)/(2)`
B. `(-a)/(2)`
C. 2a
D. a

Answer

To solve the problem, we start with the given equation and manipulate it step by step.

### Step 1: Start with the given equation
We have:
\[ x + \frac{a}{x} = 1 \]

### Step 2: Multiply through by \(x\) to eliminate the fraction
Multiplying both sides by \(x\) gives:
\[ x^2 + a = x \]

### Step 3: Rearrange the equation
Rearranging the equation, we get:
\[ x^2 - x + a = 0 \]

### Step 4: Express \(a\) in terms of \(x\)
From the rearranged equation, we can express \(a\) as:
\[ a = x - x^2 \]

### Step 5: Substitute \(a\) into the expression we need to evaluate
We need to find the value of:
\[ \frac{x^3 - x^2}{x^2 + x + a} \]
Substituting \(a\) from the previous step:
\[ \frac{x^3 - x^2}{x^2 + x + (x - x^2)} \]
This simplifies to:
\[ \frac{x^3 - x^2}{x^2 + x + x - x^2} = \frac{x^3 - x^2}{2x} \]

### Step 6: Simplify the expression
Now, we can simplify the expression:
\[ \frac{x^3 - x^2}{2x} = \frac{x^2(x - 1)}{2x} = \frac{x(x - 1)}{2} \]

### Step 7: Substitute \(x + \frac{a}{x} = 1\) to find \(x\)
From the original equation, we know:
\[ x + \frac{a}{x} = 1 \implies x + \frac{x - x^2}{x} = 1 \]
This means:
\[ x + 1 - x = 1 \]
This confirms our manipulation is consistent.

### Step 8: Final expression in terms of \(a\)
Since we have \(x(x - 1) = -a\) from our earlier manipulation, we can express our final answer as:
\[ \frac{-a}{2} \]

### Final Answer
Thus, the value of \(\frac{x^3 - x^2}{x^2 + x + a}\) in terms of \(a\) is:
\[ -\frac{a}{2} \]

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