If the roots of the equation `x^(2) - (a-1) x+ ( a+b) = 0 and ax^(2) - 2x + b= 0` are identical, then what are the values of a and b?

Options

A. `a=2,b=4`
B. `a=2,b=-4`
C. `a=1,b=1/2`
D. `a=-1,b=-1/2`

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) such that the roots of the two given quadratic equations are identical.

### Step 1: Write down the equations
The two equations given are:
1. \( x^2 - (a - 1)x + (a + b) = 0 \)
2. \( ax^2 - 2x + b = 0 \)

### Step 2: Identify the sum and product of the roots
For the first equation \( x^2 - (a - 1)x + (a + b) = 0 \):
- The sum of the roots \( \alpha + \beta = a - 1 \)
- The product of the roots \( \alpha \beta = a + b \)

For the second equation \( ax^2 - 2x + b = 0 \):
- The sum of the roots \( \alpha + \beta = \frac{2}{a} \)
- The product of the roots \( \alpha \beta = \frac{b}{a} \)

### Step 3: Set the sums of the roots equal
Since the roots are identical, we can set the sums equal:
\[ a - 1 = \frac{2}{a} \]

### Step 4: Multiply through by \( a \) to eliminate the fraction
Multiplying both sides by \( a \) gives:
\[ a^2 - a = 2 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ a^2 - a - 2 = 0 \]

### Step 6: Factor the quadratic
Factoring the quadratic:
\[ (a - 2)(a + 1) = 0 \]
Thus, the solutions for \( a \) are:
\[ a = 2 \quad \text{or} \quad a = -1 \]

### Step 7: Find corresponding values of \( b \)
Now we will find \( b \) for both values of \( a \).

**Case 1: \( a = 2 \)**
Substituting \( a = 2 \) into the equation for the product of the roots:
\[ a + b = \frac{b}{a} \]
This gives:
\[ 2 + b = \frac{b}{2} \]
Multiplying through by 2:
\[ 4 + 2b = b \]
Rearranging gives:
\[ b - 2b = 4 \implies -b = 4 \implies b = -4 \]

**Case 2: \( a = -1 \)**
Substituting \( a = -1 \) into the equation for the product of the roots:
\[ a + b = \frac{b}{a} \]
This gives:
\[ -1 + b = -b \]
Rearranging gives:
\[ -1 + b + b = 0 \implies 2b = 1 \implies b = \frac{1}{2} \]

### Final Values
Thus, we have two pairs of values:
1. \( (a, b) = (2, -4) \)
2. \( (a, b) = (-1, \frac{1}{2}) \)

### Summary
The values of \( a \) and \( b \) are:
- \( a = 2, b = -4 \)
- \( a = -1, b = \frac{1}{2} \)