Questions: Subtract: 4/(x^2-9x+14)-3/(x^2-3x+2)=

Solution Steps

To subtract the given fractions, we need to find a common denominator. The denominators are quadratic expressions, so we will factor them first. After factoring, we will rewrite each fraction with the common denominator and then perform the subtraction.

We start by factoring the denominators of the fractions: \[ x^2 - 9x + 14 = (x - 7)(x - 2) \] \[ x^2 - 3x + 2 = (x - 2)(x - 1) \]

The common denominator for the two fractions is: \[ (x - 7)(x - 2)^2(x - 1) \]

We rewrite each fraction with the common denominator: \[ \frac{4}{(x - 7)(x - 2)} = \frac{4(x - 2)}{(x - 7)(x - 2)^2} \] \[ \frac{3}{(x - 2)(x - 1)} = \frac{3(x - 7)}{(x - 2)(x - 1)(x - 7)} = \frac{3(x - 7)}{(x - 7)(x - 2)(x - 1)} \]

Now we perform the subtraction: \[ \frac{4(x - 2) - 3(x - 7)}{(x - 7)(x - 2)(x - 1)} \]

Simplifying the numerator: \[ 4(x - 2) - 3(x - 7) = 4x - 8 - 3x + 21 = x + 13 \] Thus, the result is: \[ \frac{x + 13}{(x - 7)(x - 2)(x - 1)} \]

Final Answer