Solve each of the following quadratic equations: `(a)/((ax-1))+(b)/((bx-1))=(a+b),xne(1)/(a),(1)/(b)`

Explanation

To solve the quadratic equation given by:

\[ \frac{a}{ax-1} + \frac{b}{bx-1} = a + b \]

where \( x \neq \frac{1}{a} \) and \( x \neq \frac{1}{b} \), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the left-hand side is \((ax - 1)(bx - 1)\).

### Step 2: Rewrite the equation
We can rewrite the equation as follows:

\[ \frac{a(bx - 1) + b(ax - 1)}{(ax - 1)(bx - 1)} = a + b \]

### Step 3: Simplify the numerator
Expanding the numerator gives:

\[ abx - a + abx - b = 2abx - (a + b) \]

Thus, the equation becomes:

\[ \frac{2abx - (a + b)}{(ax - 1)(bx - 1)} = a + b \]

### Step 4: Cross-multiply
Cross-multiplying gives:

\[ 2abx - (a + b) = (a + b)(ax - 1)(bx - 1) \]

### Step 5: Expand the right-hand side
Expanding the right-hand side:

\[ (a + b)(abx^2 - (a + b)x + 1) \]

### Step 6: Set the equation to zero
Now we have:

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

Rearranging gives:

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

### Step 7: Combine like terms
Combining like terms results in:

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

### Step 8: Factor the quadratic equation
We can factor the quadratic equation:

\[ 0 = (a + b)(abx^2 - (a + b)x + 1) \]

### Step 9: Solve for \(x\)
Setting each factor to zero gives us:

1. \(a + b = 0\) (not applicable since \(a\) and \(b\) are non-zero)
2. \(abx^2 - (a + b)x + 1 = 0\)

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

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

### Step 10: Final roots
Thus, the roots of the quadratic equation are:

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