Simplify (2x)/(x^2-6x+9)+4/(x^2+2x-15)

$\frac{2 x}{x^{2} - 6 x + 9} + \frac{4}{x^{2} + 2 x - 15}$

Step 1

Simplify each term.

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{2 x}{x^{2} - 6 x + 3^{2}} + \frac{4}{x^{2} + 2 x - 15}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Rewrite the polynomial.

$\frac{2 x}{x^{2} - 2 \cdot x \cdot 3 + 3^{2}} + \frac{4}{x^{2} + 2 x - 15}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{2 x}{{(x - 3)}^{2}} + \frac{4}{x^{2} + 2 x - 15}$

Factor $x^{2} + 2 x - 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $2$ .

$- 3, 5$

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

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

Multiply $\frac{2 x}{{(x - 3)}^{2}}$ by $\frac{x + 5}{x + 5}$ .

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

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

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

Raise $x - 3$ to the power of $1$ .

$\frac{2 x (x + 5)}{{(x - 3)}^{2} (x + 5)} + \frac{4 (x - 3)}{{(x - 3)}^{1} (x - 3) (x + 5)}$

Raise $x - 3$ to the power of $1$ .

$\frac{2 x (x + 5)}{{(x - 3)}^{2} (x + 5)} + \frac{4 (x - 3)}{{(x - 3)}^{1} {(x - 3)}^{1} (x + 5)}$

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

$\frac{2 x (x + 5)}{{(x - 3)}^{2} (x + 5)} + \frac{4 (x - 3)}{{(x - 3)}^{1 + 1} (x + 5)}$

Add $1$ and $1$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Factor $2$ out of $2 x (x + 5) + 4 (x - 3)$ .

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

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

Factor $2$ out of $4 (x - 3)$ .

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

Factor $2$ out of $2 (x (x + 5)) + 2 (2 (x - 3))$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Move $5$ to the left of $x$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 3$ .

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

Add $5 x$ and $2 x$ .

$\frac{2 (x^{2} + 7 x - 6)}{{(x - 3)}^{2} (x + 5)}$

Simplify (2x)/(x^2-6x+9)+4/(x^2+2x-15)

Related Questions

Support LCMGCF.com by disabling your adblocker.