Question

simplify the expression: \\(\frac{3x}{x^2 - 7x + 10} - \frac{5}{x^2 - 9x + 20}\\)

Explanation:

Step1: Factor denominators

Factor quadratic expressions:
$x^2-7x+10=(x-2)(x-5)$
$x^2-9x+20=(x-4)(x-5)$
Rewrite the expression:
$\frac{3x}{(x-2)(x-5)} - \frac{5}{(x-4)(x-5)}$

Step2: Find common denominator

Common denominator is $(x-2)(x-4)(x-5)$
Adjust numerators:
$\frac{3x(x-4)}{(x-2)(x-4)(x-5)} - \frac{5(x-2)}{(x-2)(x-4)(x-5)}$

Step3: Combine numerators

Subtract the numerators over common denominator:
$\frac{3x(x-4)-5(x-2)}{(x-2)(x-4)(x-5)}$

Step4: Expand numerator terms

Calculate expanded numerator:
$3x(x-4)=3x^2-12x$
$5(x-2)=5x-10$
$\frac{3x^2-12x-5x+10}{(x-2)(x-4)(x-5)}$

Step5: Simplify numerator

Combine like terms in numerator:
$\frac{3x^2-17x+10}{(x-2)(x-4)(x-5)}$

Step6: Factor simplified numerator

Factor the quadratic numerator:
$3x^2-17x+10=(3x-2)(x-5)$
Rewrite expression:
$\frac{(3x-2)(x-5)}{(x-2)(x-4)(x-5)}$

Step7: Cancel common factors

Cancel $(x-5)$ (where $x
eq5$):
$\frac{3x-2}{(x-2)(x-4)}$

Step8: Expand denominator (optional)

Expand denominator for standard form:
$(x-2)(x-4)=x^2-6x+8$
$\frac{3x-2}{x^2-6x+8}$

Answer:

$\frac{3x-2}{(x-2)(x-4)}$ (or $\frac{3x-2}{x^2-6x+8}$), where $x
eq2,4,5$