Simplify (4a^2-36a)/(2a^4-24a^3+54a^2)

$\frac{4 a^{2} - 36 a}{2 a^{4} - 24 a^{3} + 54 a^{2}}$

Step 1

Factor $4 a$ out of $4 a^{2} - 36 a$ .

Factor $4 a$ out of $4 a^{2}$ .

$\frac{4 a (a) - 36 a}{2 a^{4} - 24 a^{3} + 54 a^{2}}$

Factor $4 a$ out of $- 36 a$ .

$\frac{4 a (a) + 4 a (- 9)}{2 a^{4} - 24 a^{3} + 54 a^{2}}$

Factor $4 a$ out of $4 a (a) + 4 a (- 9)$ .

$\frac{4 a (a - 9)}{2 a^{4} - 24 a^{3} + 54 a^{2}}$

Step 2

Simplify the denominator.

Factor $2 a^{2}$ out of $2 a^{4} - 24 a^{3} + 54 a^{2}$ .

Factor $2 a^{2}$ out of $2 a^{4}$ .

$\frac{4 a (a - 9)}{2 a^{2} (a^{2}) - 24 a^{3} + 54 a^{2}}$

Factor $2 a^{2}$ out of $- 24 a^{3}$ .

$\frac{4 a (a - 9)}{2 a^{2} (a^{2}) + 2 a^{2} (- 12 a) + 54 a^{2}}$

Factor $2 a^{2}$ out of $54 a^{2}$ .

$\frac{4 a (a - 9)}{2 a^{2} (a^{2}) + 2 a^{2} (- 12 a) + 2 a^{2} (27)}$

Factor $2 a^{2}$ out of $2 a^{2} (a^{2}) + 2 a^{2} (- 12 a)$ .

$\frac{4 a (a - 9)}{2 a^{2} (a^{2} - 12 a) + 2 a^{2} (27)}$

Factor $2 a^{2}$ out of $2 a^{2} (a^{2} - 12 a) + 2 a^{2} (27)$ .

$\frac{4 a (a - 9)}{2 a^{2} (a^{2} - 12 a + 27)}$

Factor $a^{2} - 12 a + 27$ 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 $27$ and whose sum is $- 12$ .

$- 9, - 3$

Write the factored form using these integers.

$\frac{4 a (a - 9)}{2 a^{2} ((a - 9) (a - 3))}$

$\frac{4 a (a - 9)}{2 a^{2} (a - 9) (a - 3)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 a (a - 9)$ .

$\frac{2 (2 a (a - 9))}{2 a^{2} (a - 9) (a - 3)}$

Cancel the common factors.

Factor $2$ out of $2 a^{2} (a - 9) (a - 3)$ .

$\frac{2 (2 a (a - 9))}{2 ((a^{2} (a - 9)) (a - 3))}$

Cancel the common factor.

$\frac{2 (2 a (a - 9))}{2 ((a^{2} (a - 9)) (a - 3))}$

Rewrite the expression.

$\frac{2 a (a - 9)}{(a^{2} (a - 9)) (a - 3)}$

Cancel the common factor of $a$ and $a^{2}$ .

Factor $a$ out of $2 a (a - 9)$ .

$\frac{a (2 (a - 9))}{(a^{2} (a - 9)) (a - 3)}$

Cancel the common factors.

Factor $a$ out of $(a^{2} (a - 9)) (a - 3)$ .

$\frac{a (2 (a - 9))}{a ((a (a - 9)) (a - 3))}$

Cancel the common factor.

$\frac{a (2 (a - 9))}{a ((a (a - 9)) (a - 3))}$

Rewrite the expression.

$\frac{2 (a - 9)}{(a (a - 9)) (a - 3)}$

Cancel the common factor of $a - 9$ .

Cancel the common factor.

$\frac{2 (a - 9)}{a (a - 9) (a - 3)}$

Rewrite the expression.

$\frac{2}{(a) (a - 3)}$

$\frac{2}{a (a - 3)}$

Simplify (4a^2-36a)/(2a^4-24a^3+54a^2)

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