`4/(x-3)+2/(x-2)=2`
If the solution of the equation above in simplest radical form are `x=apmsqrt(b)`, what are the values of a and b?

Options

A. `a=4, b=3`
B. `a=-4, b=5`
C. `a=3, b=5`
D. `a=-3, b=5`

Answer

To solve the equation \( \frac{4}{x-3} + \frac{2}{x-2} = 2 \), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
The denominators are \( x-3 \) and \( x-2 \). The LCM of these two denominators is \( (x-3)(x-2) \).

### Step 2: Rewrite the Equation
Multiply both sides of the equation by the LCM to eliminate the denominators:
\[ 4(x-2) + 2(x-3) = 2(x-3)(x-2) \]

### Step 3: Expand Both Sides
Now we will expand both sides:
- Left side:
\[ 4(x-2) + 2(x-3) = 4x - 8 + 2x - 6 = 6x - 14 \]
- Right side:
\[ 2(x-3)(x-2) = 2(x^2 - 5x + 6) = 2x^2 - 10x + 12 \]

### Step 4: Set the Equation to Zero
Now we set the equation to zero:
\[ 6x - 14 = 2x^2 - 10x + 12 \]
Rearranging gives:
\[ 0 = 2x^2 - 10x + 12 - 6x + 14 \]
Combining like terms:
\[ 0 = 2x^2 - 16x + 26 \]

### Step 5: Simplify the Equation
We can divide the entire equation by 2 to simplify:
\[ 0 = x^2 - 8x + 13 \]

### Step 6: Use the Quadratic Formula
Now we will apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1 \), \( b = -8 \), and \( c = 13 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-8)^2 - 4(1)(13) = 64 - 52 = 12 \]

Now substituting into the quadratic formula:
\[ x = \frac{-(-8) \pm \sqrt{12}}{2(1)} = \frac{8 \pm \sqrt{12}}{2} \]

### Step 7: Simplify the Result
We can simplify \( \sqrt{12} \) as follows:
\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]
Thus, we have:
\[ x = \frac{8 \pm 2\sqrt{3}}{2} = 4 \pm \sqrt{3} \]

### Final Result
The solutions are \( x = 4 + \sqrt{3} \) and \( x = 4 - \sqrt{3} \).

In the form \( x = a \pm \sqrt{b} \), we identify:
- \( a = 4 \)
- \( b = 3 \)

### Conclusion
The values of \( a \) and \( b \) are:
- \( a = 4 \)
- \( b = 3 \)