Simplify 3/(2x-5)+21/(8x^2-14x-15)

$\frac{3}{2 x - 5} + \frac{21}{8 x^{2} - 14 x - 15}$

Step 1

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 $a \cdot c = 8 \cdot - 15 = - 120$ and whose sum is $b = - 14$ .

Factor $- 14$ out of $- 14 x$ .

$\frac{3}{2 x - 5} + \frac{21}{8 x^{2} - 14 (x) - 15}$

Rewrite $- 14$ as $6$ plus $- 20$

$\frac{3}{2 x - 5} + \frac{21}{8 x^{2} + (6 - 20) x - 15}$

Apply the distributive property.

$\frac{3}{2 x - 5} + \frac{21}{8 x^{2} + 6 x - 20 x - 15}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{3}{2 x - 5} + \frac{21}{(8 x^{2} + 6 x) - 20 x - 15}$

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

$\frac{3}{2 x - 5} + \frac{21}{2 x (4 x + 3) - 5 (4 x + 3)}$

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

$\frac{3}{2 x - 5} + \frac{21}{(4 x + 3) (2 x - 5)}$

Step 2

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

$\frac{3}{2 x - 5} \cdot \frac{4 x + 3}{4 x + 3} + \frac{21}{(4 x + 3) (2 x - 5)}$

Step 3

Write each expression with a common denominator of $(2 x - 5) (4 x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{3 (4 x + 3)}{(2 x - 5) (4 x + 3)} + \frac{21}{(4 x + 3) (2 x - 5)}$

Reorder the factors of $(4 x + 3) (2 x - 5)$ .

$\frac{3 (4 x + 3)}{(2 x - 5) (4 x + 3)} + \frac{21}{(2 x - 5) (4 x + 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (4 x + 3) + 21}{(2 x - 5) (4 x + 3)}$

Step 5

Simplify the numerator.

Factor $3$ out of $3 (4 x + 3) + 21$ .

Factor $3$ out of $21$ .

$\frac{3 (4 x + 3) + 3 \cdot 7}{(2 x - 5) (4 x + 3)}$

Factor $3$ out of $3 (4 x + 3) + 3 \cdot 7$ .

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

Add $3$ and $7$ .

$\frac{3 (4 x + 10)}{(2 x - 5) (4 x + 3)}$

Factor $2$ out of $4 x + 10$ .

Factor $2$ out of $4 x$ .

$\frac{3 (2 (2 x) + 10)}{(2 x - 5) (4 x + 3)}$

Factor $2$ out of $10$ .

$\frac{3 (2 (2 x) + 2 (5))}{(2 x - 5) (4 x + 3)}$

Factor $2$ out of $2 (2 x) + 2 (5)$ .

$\frac{3 (2 (2 x + 5))}{(2 x - 5) (4 x + 3)}$

$\frac{3 \cdot 2 (2 x + 5)}{(2 x - 5) (4 x + 3)}$

Multiply $3$ by $2$ .

$\frac{6 (2 x + 5)}{(2 x - 5) (4 x + 3)}$

Simplify 3/(2x-5)+21/(8x^2-14x-15)

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