Solve for a (3x-5)/(4x+3)+(-2x+1)/(7x-2)=(ax^2+bx+c)/((4x+3)(7x-2))

$\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2} = \frac{a x^{2} + b x + c}{(4 x + 3) (7 x - 2)}$

Rewrite the equation as $\frac{a x^{2} + b x + c}{(4 x + 3) (7 x - 2)} = \frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}$ .

$\frac{a x^{2} + b x + c}{(4 x + 3) (7 x - 2)} = \frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}$

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

$\frac{a x^{2} + b x + c}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2)) = (\frac{3 x - 5}{4 x + 3} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

Simplify.

Simplify the left side.

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 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 $\frac{7 x - 2}{7 x - 2}$ .

$a x^{2} + b x + c = (\frac{3 x - 5}{4 x + 3} \cdot \frac{7 x - 2}{7 x - 2} + \frac{- 2 x + 1}{7 x - 2}) ((4 x + 3) (7 x - 2))$

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

$a x^{2} + b x + c = (\frac{3 x - 5}{4 x + 3} \cdot \frac{7 x - 2}{7 x - 2} + \frac{- 2 x + 1}{7 x - 2} \cdot \frac{4 x + 3}{4 x + 3}) ((4 x + 3) (7 x - 2))$

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 $\frac{7 x - 2}{7 x - 2}$ .

$a x^{2} + b x + c = (\frac{(3 x - 5) (7 x - 2)}{(4 x + 3) (7 x - 2)} + \frac{- 2 x + 1}{7 x - 2} \cdot \frac{4 x + 3}{4 x + 3}) ((4 x + 3) (7 x - 2))$

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

$a x^{2} + b x + c = (\frac{(3 x - 5) (7 x - 2)}{(4 x + 3) (7 x - 2)} + \frac{(- 2 x + 1) (4 x + 3)}{(7 x - 2) (4 x + 3)}) ((4 x + 3) (7 x - 2))$

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

$a x^{2} + b x + c = (\frac{(3 x - 5) (7 x - 2)}{(4 x + 3) (7 x - 2)} + \frac{(- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)}) ((4 x + 3) (7 x - 2))$

Combine the numerators over the common denominator.

$a x^{2} + b x + c = \frac{(3 x - 5) (7 x - 2) + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Simplify the numerator.

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

Apply the distributive property.

$a x^{2} + b x + c = \frac{3 x (7 x - 2) - 5 (7 x - 2) + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$a x^{2} + b x + c = \frac{3 x (7 x) + 3 x \cdot - 2 - 5 (7 x - 2) + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$a x^{2} + b x + c = \frac{3 x (7 x) + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$a x^{2} + b x + c = \frac{3 \cdot 7 x \cdot x + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

Move $x$ .

$a x^{2} + b x + c = \frac{3 \cdot 7 (x \cdot x) + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $x$ by $x$ .

$a x^{2} + b x + c = \frac{3 \cdot 7 x^{2} + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $3$ by $7$ .

$a x^{2} + b x + c = \frac{21 x^{2} + 3 x \cdot - 2 - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $- 2$ by $3$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 6 x - 5 (7 x) - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $7$ by $- 5$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 6 x - 35 x - 5 \cdot - 2 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $- 5$ by $- 2$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 6 x - 35 x + 10 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 + (- 2 x + 1) (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

Apply the distributive property.

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 2 x (4 x + 3) + 1 (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 2 x (4 x) - 2 x \cdot 3 + 1 (4 x + 3)}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Apply the distributive property.

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 2 x (4 x) - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 x \cdot x - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

Move $x$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 (x \cdot x) - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $x$ by $x$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 2 \cdot 4 x^{2} - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $- 2$ by $4$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 2 x \cdot 3 + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $3$ by $- 2$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 1 (4 x) + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $4 x$ by $1$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 4 x + 1 \cdot 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Multiply $3$ by $1$ .

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 6 x + 4 x + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

$a x^{2} + b x + c = \frac{21 x^{2} - 41 x + 10 - 8 x^{2} - 2 x + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

$a x^{2} + b x + c = \frac{13 x^{2} - 41 x + 10 - 2 x + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

$a x^{2} + b x + c = \frac{13 x^{2} - 43 x + 10 + 3}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Add $10$ and $3$ .

$a x^{2} + b x + c = \frac{13 x^{2} - 43 x + 13}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

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

Cancel the common factor.

$a x^{2} + b x + c = \frac{13 x^{2} - 43 x + 13}{(4 x + 3) (7 x - 2)} ((4 x + 3) (7 x - 2))$

Rewrite the expression.

$a x^{2} + b x + c = 13 x^{2} - 43 x + 13$

Solve for $a$ .

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

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

$a x^{2} + c = 13 x^{2} - 43 x + 13 - b x$

Subtract $c$ from both sides of the equation.

$a x^{2} = 13 x^{2} - 43 x + 13 - b x - c$

Divide each term in $a x^{2} = 13 x^{2} - 43 x + 13 - b x - c$ by $x^{2}$ and simplify.

Divide each term in $a x^{2} = 13 x^{2} - 43 x + 13 - b x - c$ by $x^{2}$ .

$\frac{a x^{2}}{x^{2}} = \frac{13 x^{2}}{x^{2}} + \frac{- 43 x}{x^{2}} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Simplify the left side.

Cancel the common factor of $x^{2}$ .

Cancel the common factor.

$\frac{a x^{2}}{x^{2}} = \frac{13 x^{2}}{x^{2}} + \frac{- 43 x}{x^{2}} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Divide $a$ by $1$ .

$a = \frac{13 x^{2}}{x^{2}} + \frac{- 43 x}{x^{2}} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $x^{2}$ .

Cancel the common factor.

$a = \frac{13 x^{2}}{x^{2}} + \frac{- 43 x}{x^{2}} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Divide $13$ by $1$ .

$a = 13 + \frac{- 43 x}{x^{2}} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $- 43 x$ .

$a = 13 + \frac{x \cdot - 43}{x^{2}} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Cancel the common factors.

Factor $x$ out of $x^{2}$ .

$a = 13 + \frac{x \cdot - 43}{x \cdot x} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Cancel the common factor.

$a = 13 + \frac{x \cdot - 43}{x \cdot x} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Rewrite the expression.

$a = 13 + \frac{- 43}{x} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Move the negative in front of the fraction.

$a = 13 - \frac{43}{x} + \frac{13}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $- b x$ .

$a = 13 - \frac{43}{x} + \frac{13}{x^{2}} + \frac{x (- b)}{x^{2}} + \frac{- c}{x^{2}}$

Cancel the common factors.

Factor $x$ out of $x^{2}$ .

$a = 13 - \frac{43}{x} + \frac{13}{x^{2}} + \frac{x (- b)}{x \cdot x} + \frac{- c}{x^{2}}$

Cancel the common factor.

$a = 13 - \frac{43}{x} + \frac{13}{x^{2}} + \frac{x (- b)}{x \cdot x} + \frac{- c}{x^{2}}$

Rewrite the expression.

$a = 13 - \frac{43}{x} + \frac{13}{x^{2}} + \frac{- b}{x} + \frac{- c}{x^{2}}$

Move the negative in front of the fraction.

$a = 13 - \frac{43}{x} + \frac{13}{x^{2}} - \frac{b}{x} + \frac{- c}{x^{2}}$

Move the negative in front of the fraction.

$a = 13 - \frac{43}{x} + \frac{13}{x^{2}} - \frac{b}{x} - \frac{c}{x^{2}}$

Solve for a (3x-5)/(4x+3)+(-2x+1)/(7x-2)=(ax^2+bx+c)/((4x+3)(7x-2))

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