Simplify (2x^2)/(6x^2-13x+6)+(2x-3)/(3x^2-11x+6)

$\frac{2 x^{2}}{6 x^{2} - 13 x + 6} + \frac{2 x - 3}{3 x^{2} - 11 x + 6}$

Simplify each term.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot 6 = 36$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 x$ .

$\frac{2 x^{2}}{6 x^{2} - 13 (x) + 6} + \frac{2 x - 3}{3 x^{2} - 11 x + 6}$

Rewrite $- 13$ as $- 4$ plus $- 9$

$\frac{2 x^{2}}{6 x^{2} + (- 4 - 9) x + 6} + \frac{2 x - 3}{3 x^{2} - 11 x + 6}$

Apply the distributive property.

$\frac{2 x^{2}}{6 x^{2} - 4 x - 9 x + 6} + \frac{2 x - 3}{3 x^{2} - 11 x + 6}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2 x^{2}}{(6 x^{2} - 4 x) - 9 x + 6} + \frac{2 x - 3}{3 x^{2} - 11 x + 6}$

Factor out the greatest common factor (GCF) from each group.

$\frac{2 x^{2}}{2 x (3 x - 2) - 3 (3 x - 2)} + \frac{2 x - 3}{3 x^{2} - 11 x + 6}$

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} + \frac{2 x - 3}{3 x^{2} - 11 x + 6}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 6 = 18$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} + \frac{2 x - 3}{3 x^{2} - 11 (x) + 6}$

Rewrite $- 11$ as $- 2$ plus $- 9$

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} + \frac{2 x - 3}{3 x^{2} + (- 2 - 9) x + 6}$

Apply the distributive property.

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} + \frac{2 x - 3}{3 x^{2} - 2 x - 9 x + 6}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} + \frac{2 x - 3}{(3 x^{2} - 2 x) - 9 x + 6}$

Factor out the greatest common factor (GCF) from each group.

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} + \frac{2 x - 3}{x (3 x - 2) - 3 (3 x - 2)}$

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} + \frac{2 x - 3}{(3 x - 2) (x - 3)}$

To write $\frac{2 x^{2}}{(3 x - 2) (2 x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} \cdot \frac{x - 3}{x - 3} + \frac{2 x - 3}{(3 x - 2) (x - 3)}$

To write $\frac{2 x - 3}{(3 x - 2) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{2 x - 3}{2 x - 3}$ .

$\frac{2 x^{2}}{(3 x - 2) (2 x - 3)} \cdot \frac{x - 3}{x - 3} + \frac{2 x - 3}{(3 x - 2) (x - 3)} \cdot \frac{2 x - 3}{2 x - 3}$

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

Multiply $\frac{2 x^{2}}{(3 x - 2) (2 x - 3)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{2 x^{2} (x - 3)}{(3 x - 2) (2 x - 3) (x - 3)} + \frac{2 x - 3}{(3 x - 2) (x - 3)} \cdot \frac{2 x - 3}{2 x - 3}$

Multiply $\frac{2 x - 3}{(3 x - 2) (x - 3)}$ by $\frac{2 x - 3}{2 x - 3}$ .

$\frac{2 x^{2} (x - 3)}{(3 x - 2) (2 x - 3) (x - 3)} + \frac{(2 x - 3) (2 x - 3)}{(3 x - 2) (x - 3) (2 x - 3)}$

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

$\frac{2 x^{2} (x - 3)}{(2 x - 3) (3 x - 2) (x - 3)} + \frac{(2 x - 3) (2 x - 3)}{(3 x - 2) (x - 3) (2 x - 3)}$

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

$\frac{2 x^{2} (x - 3)}{(2 x - 3) (3 x - 2) (x - 3)} + \frac{(2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{2 x^{2} (x - 3) + (2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Simplify the numerator.

Apply the distributive property.

$\frac{2 x^{2} x + 2 x^{2} \cdot - 3 + (2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 x^{2} \cdot - 3 + (2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$\frac{2 (x^{1} x^{2}) + 2 x^{2} \cdot - 3 + (2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x^{1 + 2} + 2 x^{2} \cdot - 3 + (2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 x^{2} \cdot - 3 + (2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $- 3$ by $2$ .

$\frac{2 x^{3} - 6 x^{2} + (2 x - 3) (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

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

Apply the distributive property.

$\frac{2 x^{3} - 6 x^{2} + 2 x (2 x - 3) - 3 (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Apply the distributive property.

$\frac{2 x^{3} - 6 x^{2} + 2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3)}{(2 x - 3) (3 x - 2) (x - 3)}$

Apply the distributive property.

$\frac{2 x^{3} - 6 x^{2} + 2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3}{(2 x - 3) (3 x - 2) (x - 3)}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 6 x^{2} + 2 \cdot 2 x \cdot x + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3}{(2 x - 3) (3 x - 2) (x - 3)}$

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

Move $x$ .

$\frac{2 x^{3} - 6 x^{2} + 2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 6 x^{2} + 2 \cdot 2 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $2$ by $2$ .

$\frac{2 x^{3} - 6 x^{2} + 4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $- 3$ by $2$ .

$\frac{2 x^{3} - 6 x^{2} + 4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $2$ by $- 3$ .

$\frac{2 x^{3} - 6 x^{2} + 4 x^{2} - 6 x - 6 x - 3 \cdot - 3}{(2 x - 3) (3 x - 2) (x - 3)}$

Multiply $- 3$ by $- 3$ .

$\frac{2 x^{3} - 6 x^{2} + 4 x^{2} - 6 x - 6 x + 9}{(2 x - 3) (3 x - 2) (x - 3)}$

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

$\frac{2 x^{3} - 6 x^{2} + 4 x^{2} - 12 x + 9}{(2 x - 3) (3 x - 2) (x - 3)}$

Add $- 6 x^{2}$ and $4 x^{2}$ .

$\frac{2 x^{3} - 2 x^{2} - 12 x + 9}{(2 x - 3) (3 x - 2) (x - 3)}$

Simplify (2x^2)/(6x^2-13x+6)+(2x-3)/(3x^2-11x+6)

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