Simplify (2(x-1))/(4x^2-3x-1)+(x+2)/(4x^2+7x-2)

$\frac{2 (x - 1)}{4 x^{2} - 3 x - 1} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

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 $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

$\frac{2 (x - 1)}{4 x^{2} - 3 (x) - 1} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

Rewrite $- 3$ as $1$ plus $- 4$

$\frac{2 (x - 1)}{4 x^{2} + (1 - 4) x - 1} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

Apply the distributive property.

$\frac{2 (x - 1)}{4 x^{2} + 1 x - 4 x - 1} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

Multiply $x$ by $1$ .

$\frac{2 (x - 1)}{4 x^{2} + x - 4 x - 1} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2 (x - 1)}{(4 x^{2} + x) - 4 x - 1} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

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

$\frac{2 (x - 1)}{x (4 x + 1) - (4 x + 1)} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$\frac{2 (x - 1)}{(4 x + 1) (x - 1)} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{2 (x - 1)}{(4 x + 1) (x - 1)} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

Rewrite the expression.

$\frac{2}{4 x + 1} + \frac{x + 2}{4 x^{2} + 7 x - 2}$

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 $a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$\frac{2}{4 x + 1} + \frac{x + 2}{4 x^{2} + 7 (x) - 2}$

Rewrite $7$ as $- 1$ plus $8$

$\frac{2}{4 x + 1} + \frac{x + 2}{4 x^{2} + (- 1 + 8) x - 2}$

Apply the distributive property.

$\frac{2}{4 x + 1} + \frac{x + 2}{4 x^{2} - 1 x + 8 x - 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2}{4 x + 1} + \frac{x + 2}{(4 x^{2} - 1 x) + 8 x - 2}$

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

$\frac{2}{4 x + 1} + \frac{x + 2}{x (4 x - 1) + 2 (4 x - 1)}$

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$\frac{2}{4 x + 1} + \frac{x + 2}{(4 x - 1) (x + 2)}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{2}{4 x + 1} + \frac{x + 2}{(4 x - 1) (x + 2)}$

Rewrite the expression.

$\frac{2}{4 x + 1} + \frac{1}{4 x - 1}$

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

$\frac{2}{4 x + 1} \cdot \frac{4 x - 1}{4 x - 1} + \frac{1}{4 x - 1}$

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

$\frac{2}{4 x + 1} \cdot \frac{4 x - 1}{4 x - 1} + \frac{1}{4 x - 1} \cdot \frac{4 x + 1}{4 x + 1}$

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

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

$\frac{2 (4 x - 1)}{(4 x + 1) (4 x - 1)} + \frac{1}{4 x - 1} \cdot \frac{4 x + 1}{4 x + 1}$

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

$\frac{2 (4 x - 1)}{(4 x + 1) (4 x - 1)} + \frac{4 x + 1}{(4 x - 1) (4 x + 1)}$

Reorder the factors of $(4 x - 1) (4 x + 1)$ .

$\frac{2 (4 x - 1)}{(4 x + 1) (4 x - 1)} + \frac{4 x + 1}{(4 x + 1) (4 x - 1)}$

Combine the numerators over the common denominator.

$\frac{2 (4 x - 1) + 4 x + 1}{(4 x + 1) (4 x - 1)}$

Simplify the numerator.

Apply the distributive property.

$\frac{2 (4 x) + 2 \cdot - 1 + 4 x + 1}{(4 x + 1) (4 x - 1)}$

Multiply $4$ by $2$ .

$\frac{8 x + 2 \cdot - 1 + 4 x + 1}{(4 x + 1) (4 x - 1)}$

Multiply $2$ by $- 1$ .

$\frac{8 x - 2 + 4 x + 1}{(4 x + 1) (4 x - 1)}$

Add $8 x$ and $4 x$ .

$\frac{12 x - 2 + 1}{(4 x + 1) (4 x - 1)}$

Add $- 2$ and $1$ .

$\frac{12 x - 1}{(4 x + 1) (4 x - 1)}$

Simplify (2(x-1))/(4x^2-3x-1)+(x+2)/(4x^2+7x-2)

Related Questions

Support LCMGCF.com by disabling your adblocker.