Simplify (x^2-4x-12)/(x^3-10x^2+24x)

$\frac{x^{2} - 4 x - 12}{x^{3} - 10 x^{2} + 24 x}$

Step 1

Factor $x^{2} - 4 x - 12$ 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 $- 4$ .

$- 6, 2$

Write the factored form using these integers.

$\frac{(x - 6) (x + 2)}{x^{3} - 10 x^{2} + 24 x}$

Step 2

Simplify the denominator.

Factor $x$ out of $x^{3} - 10 x^{2} + 24 x$ .

Factor $x$ out of $x^{3}$ .

$\frac{(x - 6) (x + 2)}{x \cdot x^{2} - 10 x^{2} + 24 x}$

Factor $x$ out of $- 10 x^{2}$ .

$\frac{(x - 6) (x + 2)}{x \cdot x^{2} + x (- 10 x) + 24 x}$

Factor $x$ out of $24 x$ .

$\frac{(x - 6) (x + 2)}{x \cdot x^{2} + x (- 10 x) + x \cdot 24}$

Factor $x$ out of $x \cdot x^{2} + x (- 10 x)$ .

$\frac{(x - 6) (x + 2)}{x (x^{2} - 10 x) + x \cdot 24}$

Factor $x$ out of $x (x^{2} - 10 x) + x \cdot 24$ .

$\frac{(x - 6) (x + 2)}{x (x^{2} - 10 x + 24)}$

Factor $x^{2} - 10 x + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Write the factored form using these integers.

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

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

Step 3

Cancel the common factor of $x - 6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (x^2-4x-12)/(x^3-10x^2+24x)

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