Simplify ((x^2-11x+30)/(x-6))÷((x^2-12x+35)/(9x^3))

$\frac{x^{2} - 11 x + 30}{x - 6} \div \frac{x^{2} - 12 x + 35}{9 x^{3}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 11 x + 30}{x - 6} \cdot \frac{9 x^{3}}{x^{2} - 12 x + 35}$

Step 2

Factor $x^{2} - 11 x + 30$ 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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Write the factored form using these integers.

$\frac{(x - 6) (x - 5)}{x - 6} \cdot \frac{9 x^{3}}{x^{2} - 12 x + 35}$

Step 3

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{(x - 6) (x - 5)}{x - 6} \cdot \frac{9 x^{3}}{x^{2} - 12 x + 35}$

Divide $x - 5$ by $1$ .

$(x - 5) \frac{9 x^{3}}{x^{2} - 12 x + 35}$

Step 4

Factor $x^{2} - 12 x + 35$ 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 $35$ and whose sum is $- 12$ .

$- 7, - 5$

Write the factored form using these integers.

$(x - 5) \frac{9 x^{3}}{(x - 7) (x - 5)}$

Step 5

Cancel the common factor of $x - 5$ .

Factor $x - 5$ out of $(x - 7) (x - 5)$ .

$(x - 5) \frac{9 x^{3}}{(x - 5) (x - 7)}$

Cancel the common factor.

$(x - 5) \frac{9 x^{3}}{(x - 5) (x - 7)}$

Rewrite the expression.

$\frac{9 x^{3}}{x - 7}$

Simplify ((x^2-11x+30)/(x-6))÷((x^2-12x+35)/(9x^3))

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