If one of the roots of the equation `x^(2) + ax - b = 0` is, 1 then what is the value of (a - b)

Options

A. -1
B. 1
C. 2
D. -2

Answer

To solve the problem step by step, we will use the information provided about the quadratic equation and its roots.

### Step-by-Step Solution:

1. **Identify the given quadratic equation**:
The equation is given as \( x^2 + ax - b = 0 \).

2. **Use the information about the roots**:
We know that one of the roots (let's denote it as \( \beta \)) is 1. Thus, we can write:
\[ \beta = 1 \]

3. **Let the other root be \( \alpha \)**:
Now, we have two roots: \( \alpha \) and \( \beta \) (where \( \beta = 1 \)).

4. **Use the sum of the roots formula**:
The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \text{Sum of roots} = -\frac{b}{a} \]
For our equation, this translates to:
\[ \alpha + 1 = -a \]
Rearranging gives us:
\[ \alpha = -a - 1 \quad \text{(Equation 1)} \]

5. **Use the product of the roots formula**:
The product of the roots is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
For our equation, this translates to:
\[ \alpha \cdot 1 = -b \]
Thus, we have:
\[ \alpha = -b \quad \text{(Equation 2)} \]

6. **Equate the two expressions for \( \alpha \)**:
From Equation 1 and Equation 2, we have:
\[ -b = -a - 1 \]
Rearranging this gives:
\[ b = a + 1 \]

7. **Find \( a - b \)**:
We need to find the value of \( a - b \):
\[ a - b = a - (a + 1) = a - a - 1 = -1 \]

### Final Answer:
Thus, the value of \( a - b \) is \( -1 \).

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