Solve each of the following quadratic equations:
`(a)/((x-b))+(b)/((x-a))=2,xneb,a`

Answer

To solve the quadratic equation given by:

\[ \frac{a}{x-b} + \frac{b}{x-a} = 2 \]

we will follow these steps:

### Step 1: Combine the fractions
First, we need to combine the fractions on the left-hand side. The common denominator will be \((x - a)(x - b)\).

\[ \frac{a(x - a) + b(x - b)}{(x - a)(x - b)} = 2 \]

### Step 2: Expand the numerator
Now, we expand the numerator:

\[ a(x - a) + b(x - b) = ax - a^2 + bx - b^2 = (a + b)x - (a^2 + b^2) \]

So, we rewrite the equation as:

\[ \frac{(a + b)x - (a^2 + b^2)}{(x - a)(x - b)} = 2 \]

### Step 3: Cross-multiply
Next, we cross-multiply to eliminate the fraction:

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

### Step 4: Expand the right-hand side
Now we expand the right-hand side:

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

### Step 5: Rearrange the equation
Now, we equate both sides:

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

Rearranging gives us:

\[ 2x^2 - (3(a + b))x + (2ab + (a^2 + b^2)) = 0 \]

### Step 6: Simplify the equation
We can rewrite the equation as:

\[ 2x^2 - 3(a + b)x + (a^2 + b^2 + 2ab) = 0 \]

Recognizing that \(a^2 + b^2 + 2ab = (a + b)^2\), we have:

\[ 2x^2 - 3(a + b)x + (a + b)^2 = 0 \]

### Step 7: Apply the quadratic formula
Now, we will use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) where \(D\) is the discriminant.

Here, \(a = 2\), \(b = -3(a + b)\), and \(c = (a + b)^2\).

Calculating the discriminant \(D\):

\[ D = (-3(a + b))^2 - 4 \cdot 2 \cdot (a + b)^2 = 9(a + b)^2 - 8(a + b)^2 = (a + b)^2 \]

### Step 8: Find the values of \(x\)
Now substituting back into the quadratic formula:

\[ x = \frac{3(a + b) \pm \sqrt{(a + b)^2}}{2 \cdot 2} = \frac{3(a + b) \pm (a + b)}{4} \]

This gives us two solutions:

1. For the positive sign:
\[ x = \frac{3(a + b) + (a + b)}{4} = \frac{4(a + b)}{4} = a + b \]

2. For the negative sign:
\[ x = \frac{3(a + b) - (a + b)}{4} = \frac{2(a + b)}{4} = \frac{a + b}{2} \]

### Final Answer
Thus, the solutions to the equation are:

\[ x = a + b \quad \text{and} \quad x = \frac{a + b}{2} \]