Question

fill in the blank 1 point determine the correct values for a, b and c for the problem below. \\(\frac{3x - 5}{4x + 3} + \frac{-2x + 1}{7x - 2} = \frac{ax^2 + bx + c}{(4x + 3)(7x - 2)}\\)

Explanation:

Step1: Cross-multiply to eliminate denominators

Multiply both sides by $(4x+3)(7x-2)$:
$$(3x-5)(7x-2) + (-2x+1)(4x+3) = Ax^2 + Bx + C$$

Step2: Expand each product

First product: $(3x-5)(7x-2) = 21x^2 -6x -35x +10 = 21x^2 -41x +10$
Second product: $(-2x+1)(4x+3) = -8x^2 -6x +4x +3 = -8x^2 -2x +3$

Step3: Combine like terms

Add the two expanded expressions:
$$(21x^2 -8x^2) + (-41x -2x) + (10+3) = Ax^2 + Bx + C$$
$$13x^2 -43x +13 = Ax^2 + Bx + C$$

Step4: Match coefficients

Equate coefficients of corresponding terms.

Answer:

$A=13$, $B=-43$, $C=13$