Simplify (2x^2)/(x^2-7x+12)-(x^2+9)/(x^2-7x+12)

$\frac{2 x^{2}}{x^{2} - 7 x + 12} - \frac{x^{2} + 9}{x^{2} - 7 x + 12}$

Step 1

Simplify terms.

Combine the numerators over the common denominator.

$\frac{2 x^{2} - (x^{2} + 9)}{x^{2} - 7 x + 12}$

Simplify each term.

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - 1 \cdot 9}{x^{2} - 7 x + 12}$

Multiply $- 1$ by $9$ .

$\frac{2 x^{2} - x^{2} - 9}{x^{2} - 7 x + 12}$

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} - 9}{x^{2} - 7 x + 12}$

Step 2

Simplify the numerator.

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

$\frac{x^{2} - 3^{2}}{x^{2} - 7 x + 12}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{(x + 3) (x - 3)}{x^{2} - 7 x + 12}$

Step 3

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

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

Step 4

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 3}{x - 4}$

Simplify (2x^2)/(x^2-7x+12)-(x^2+9)/(x^2-7x+12)

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