Simplify (5x+7)/(x-4)-(3x)/(x^2-9x+20)

$\frac{5 x + 7}{x - 4} - \frac{3 x}{x^{2} - 9 x + 20}$

Step 1

Factor $x^{2} - 9 x + 20$ 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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Write the factored form using these integers.

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

Step 2

To write $\frac{5 x + 7}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Step 3

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

Multiply $\frac{5 x + 7}{x - 4}$ by $\frac{x - 5}{x - 5}$ .

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 5$ by $5$ .

$\frac{5 x^{2} - 25 x + 7 x + 7 \cdot - 5 - 3 x}{(x - 4) (x - 5)}$

Multiply $7$ by $- 5$ .

$\frac{5 x^{2} - 25 x + 7 x - 35 - 3 x}{(x - 4) (x - 5)}$

Add $- 25 x$ and $7 x$ .

$\frac{5 x^{2} - 18 x - 35 - 3 x}{(x - 4) (x - 5)}$

Subtract $3 x$ from $- 18 x$ .

$\frac{5 x^{2} - 21 x - 35}{(x - 4) (x - 5)}$

Simplify (5x+7)/(x-4)-(3x)/(x^2-9x+20)

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