Solve for C (3x-5)/(4x+3)+(-2x+1)/(7x-2)=(Ax^2+Bx+C)/((4x+3)(7x-2))

$\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2} = \frac{A x^{2} + B x + C}{(4 x + 3) (7 x - 2)}$

Step 1

Rewrite the equation as $\frac{A x^{2} + B x + C}{(4 x + 3) (7 x - 2)} = \frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}$ .

$\frac{A x^{2} + B x + C}{(4 x + 3) (7 x - 2)} = \frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}$

Step 2

Multiply both sides by $(4 x + 3) (7 x - 2)$ .

$\frac{A x^{2} + B x + C}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2)) = (\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

Step 3

Simplify.

Simplify the left side.

Simplify $\frac{A x^{2} + B x + C}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$ .

Cancel the common factor of $(4 x + 3) (7 x - 2)$ .

Cancel the common factor.

$\frac{A x^{2} + B x + C}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2)) = (\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

Rewrite the expression.

$A x^{2} + B x + C = (\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

Simplify with commuting.

Reorder $A$ and $x^{2}$ .

$x^{2} A + B x + C = (\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

Reorder $B$ and $x$ .

$x^{2} A + x B + C = (\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

Simplify the right side.

Simplify $(\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$ .

To write $\frac{3 x - 5}{4 x + 3}$ as a fraction with a common denominator, multiply by $\frac{7 x - 2}{7 x - 2}$ .

$x^{2} A + x B + C = (\frac{3 x - 5}{4 x + 3} \cdot \frac{7 x - 2}{7 x - 2} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

To write $\frac{- 2 x + 1}{7 x - 2}$ as a fraction with a common denominator, multiply by $\frac{4 x + 3}{4 x + 3}$ .

$x^{2} A + x B + C = (\frac{3 x - 5}{4 x + 3} \cdot \frac{7 x - 2}{7 x - 2} + \frac{- 2 x + 1}{7 x - 2} \cdot \frac{4 x + 3}{4 x + 3}) ((4 x + 3) (7 x - 2))$

Write each expression with a common denominator of $(4 x + 3) (7 x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 x - 5}{4 x + 3}$ by $\frac{7 x - 2}{7 x - 2}$ .

$x^{2} A + x B + C = (\frac{(3 x - 5) (7 x - 2)}{(4 x + 3) (7 x - 2)} + \frac{- 2 x + 1}{7 x - 2} \cdot \frac{4 x + 3}{4 x + 3}) ((4 x + 3) (7 x - 2))$

Multiply $\frac{- 2 x + 1}{7 x - 2}$ by $\frac{4 x + 3}{4 x + 3}$ .

$x^{2} A + x B + C = (\frac{(3 x - 5) (7 x - 2)}{(4 x + 3) (7 x - 2)} + \frac{(- 2 x + 1) (4 x + 3)}{(7 x - 2) (4 x + 3)}) ((4 x + 3) (7 x - 2))$

Reorder the factors of $(7 x - 2) (4 x + 3)$ .

$x^{2} A + x B + C = (\frac{(3 x - 5) (7 x - 2)}{(4 x + 3) (7 x - 2)} + \frac{(- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)}) ((4 x + 3) (7 x - 2))$

Combine the numerators over the common denominator.

$x^{2} A + x B + C = \frac{(3 x - 5) (7 x - 2) + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Simplify the numerator.

Expand $(3 x - 5) (7 x - 2)$ using the FOIL Method.

Apply the distributive property.

$x^{2} A + x B + C = \frac{3 x (7 x - 2) - 5 (7 x - 2) + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$x^{2} A + x B + C = \frac{3 x (7 x) + 3 x \cdot - 2 - 5 (7 x - 2) + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$x^{2} A + x B + C = \frac{3 x (7 x) + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{2} A + x B + C = \frac{3 \cdot 7 x \cdot x + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{2} A + x B + C = \frac{3 \cdot 7 (x \cdot x) + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $x$ by $x$ .

$x^{2} A + x B + C = \frac{3 \cdot 7 x^{2} + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $3$ by $7$ .

$x^{2} A + x B + C = \frac{21 x^{2} + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $- 2$ by $3$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 6 x - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $7$ by $- 5$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 6 x - 35 x - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $- 5$ by $- 2$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 6 x - 35 x + 10 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Subtract $35 x$ from $- 6 x$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Expand $(- 2 x + 1) (4 x + 3)$ using the FOIL Method.

Apply the distributive property.

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 2 x (4 x + 3) + 1 (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 2 x (4 x) - 2 x \cdot 3 + 1 (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 2 x (4 x) - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 x \cdot x - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 (x \cdot x) - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $x$ by $x$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 x^{2} - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $- 2$ by $4$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $3$ by $- 2$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $4 x$ by $1$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 4 x + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $3$ by $1$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 4 x + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Add $- 6 x$ and $4 x$ .

$x^{2} A + x B + C = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 2 x + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Subtract $8 x^{2}$ from $21 x^{2}$ .

$x^{2} A + x B + C = \frac{13 x^{2} - 41 x + 10 - 2 x + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Subtract $2 x$ from $- 41 x$ .

$x^{2} A + x B + C = \frac{13 x^{2} - 43 x + 10 + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Add $10$ and $3$ .

$x^{2} A + x B + C = \frac{13 x^{2} - 43 x + 13}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Cancel the common factor of $(4 x + 3) (7 x - 2)$ .

Cancel the common factor.

$x^{2} A + x B + C = \frac{13 x^{2} - 43 x + 13}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Rewrite the expression.

$x^{2} A + x B + C = 13 x^{2} - 43 x + 13$

Step 4

Move all terms not containing $C$ to the right side of the equation.

Subtract $x^{2} A$ from both sides of the equation.

$x B + C = 13 x^{2} - 43 x + 13 - x^{2} A$

Subtract $x B$ from both sides of the equation.

$C = 13 x^{2} - 43 x + 13 - x^{2} A - x B$

Solve for C (3x-5)/(4x+3)+(-2x+1)/(7x-2)=(Ax^2+Bx+C)/((4x+3)(7x-2))

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