Simplify (5x+7)/(x-2)-1/(x^2-9x+14)

$\frac{5 x + 7}{x - 2} - \frac{1}{x^{2} - 9 x + 14}$

Factor $x^{2} - 9 x + 14$ 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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Write the factored form using these integers.

$\frac{5 x + 7}{x - 2} - \frac{1}{(x - 7) (x - 2)}$

To write $\frac{5 x + 7}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{5 x + 7}{x - 2} \cdot \frac{x - 7}{x - 7} - \frac{1}{(x - 7) (x - 2)}$

Write each expression with a common denominator of $(x - 2) (x - 7)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 x + 7}{x - 2}$ by $\frac{x - 7}{x - 7}$ .

$\frac{(5 x + 7) (x - 7)}{(x - 2) (x - 7)} - \frac{1}{(x - 7) (x - 2)}$

Reorder the factors of $(x - 7) (x - 2)$ .

$\frac{(5 x + 7) (x - 7)}{(x - 2) (x - 7)} - \frac{1}{(x - 2) (x - 7)}$

Combine the numerators over the common denominator.

$\frac{(5 x + 7) (x - 7) - 1}{(x - 2) (x - 7)}$

Simplify the numerator.

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

Apply the distributive property.

$\frac{5 x (x - 7) + 7 (x - 7) - 1}{(x - 2) (x - 7)}$

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot - 7 + 7 (x - 7) - 1}{(x - 2) (x - 7)}$

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot - 7 + 7 x + 7 \cdot - 7 - 1}{(x - 2) (x - 7)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{5 (x \cdot x) + 5 x \cdot - 7 + 7 x + 7 \cdot - 7 - 1}{(x - 2) (x - 7)}$

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 5 x \cdot - 7 + 7 x + 7 \cdot - 7 - 1}{(x - 2) (x - 7)}$

Multiply $- 7$ by $5$ .

$\frac{5 x^{2} - 35 x + 7 x + 7 \cdot - 7 - 1}{(x - 2) (x - 7)}$

Multiply $7$ by $- 7$ .

$\frac{5 x^{2} - 35 x + 7 x - 49 - 1}{(x - 2) (x - 7)}$

Add $- 35 x$ and $7 x$ .

$\frac{5 x^{2} - 28 x - 49 - 1}{(x - 2) (x - 7)}$

Subtract $1$ from $- 49$ .

$\frac{5 x^{2} - 28 x - 50}{(x - 2) (x - 7)}$

Simplify (5x+7)/(x-2)-1/(x^2-9x+14)

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