Pre-Algebra Examples

$\frac{12 m n + 32 n^{2} - 54 m^{2}}{8 n - 9 m}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor $2$ out of $12 m n + 32 n^{2} - 54 m^{2}$ .

Tap for more steps...

Step 1.1.1

Factor $2$ out of $12 m n$ .

$\frac{2 (6 m n) + 32 n^{2} - 54 m^{2}}{8 n - 9 m}$

Step 1.1.2

Factor $2$ out of $32 n^{2}$ .

$\frac{2 (6 m n) + 2 (16 n^{2}) - 54 m^{2}}{8 n - 9 m}$

Step 1.1.3

Factor $2$ out of $- 54 m^{2}$ .

$\frac{2 (6 m n) + 2 (16 n^{2}) + 2 (- 27 m^{2})}{8 n - 9 m}$

Step 1.1.4

Factor $2$ out of $2 (6 m n) + 2 (16 n^{2})$ .

$\frac{2 (6 m n + 16 n^{2}) + 2 (- 27 m^{2})}{8 n - 9 m}$

Step 1.1.5

Factor $2$ out of $2 (6 m n + 16 n^{2}) + 2 (- 27 m^{2})$ .

$\frac{2 (6 m n + 16 n^{2} - 27 m^{2})}{8 n - 9 m}$

Step 1.2

Factor by grouping.

Tap for more steps...

Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 27 \cdot 16 = - 432$ and whose sum is $b = 6$ .

Tap for more steps...

Step 1.2.1.1

Reorder terms.

$\frac{2 (- 27 m^{2} + 16 n^{2} + 6 m n)}{8 n - 9 m}$

Step 1.2.1.2

Reorder $16 n^{2}$ and $6 m n$ .

$\frac{2 (- 27 m^{2} + 6 m n + 16 n^{2})}{8 n - 9 m}$

Step 1.2.1.3

Factor $6$ out of $6 m n$ .

$\frac{2 (- 27 m^{2} + 6 (m n) + 16 n^{2})}{8 n - 9 m}$

Step 1.2.1.4

Rewrite $6$ as $- 18$ plus $24$

$\frac{2 (- 27 m^{2} + (- 18 + 24) (m n) + 16 n^{2})}{8 n - 9 m}$

Step 1.2.1.5

Apply the distributive property.

$\frac{2 (- 27 m^{2} - 18 (m n) + 24 (m n) + 16 n^{2})}{8 n - 9 m}$

Step 1.2.1.6

Move parentheses.

$\frac{2 (- 27 m^{2} - 18 m n + 24 m n + 16 n^{2})}{8 n - 9 m}$

Step 1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((- 27 m^{2} - 18 m n) + 24 m n + 16 n^{2})}{8 n - 9 m}$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (9 m (- 3 m - 2 n) - 8 n (- 3 m - 2 n))}{8 n - 9 m}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $- 3 m - 2 n$ .

$\frac{2 ((- 3 m - 2 n) (9 m - 8 n))}{8 n - 9 m}$

$\frac{2 (- 3 m - 2 n) (9 m - 8 n)}{8 n - 9 m}$

Step 2

Cancel the common factor of $9 m - 8 n$ and $8 n - 9 m$ .

Tap for more steps...

Step 2.1

Factor $- 1$ out of $9 m$ .

$\frac{2 (- 3 m - 2 n) (- (- 9 m) - 8 n)}{8 n - 9 m}$

Step 2.2

Factor $- 1$ out of $- 8 n$ .

$\frac{2 (- 3 m - 2 n) (- (- 9 m) - (8 n))}{8 n - 9 m}$

Step 2.3

Factor $- 1$ out of $- (- 9 m) - (8 n)$ .

$\frac{2 (- 3 m - 2 n) (- (- 9 m + 8 n))}{8 n - 9 m}$

Step 2.4

Rewrite $- (- 9 m + 8 n)$ as $- 1 (- 9 m + 8 n)$ .

$\frac{2 (- 3 m - 2 n) (- 1 (- 9 m + 8 n))}{8 n - 9 m}$

Step 2.5

Reorder terms.

$\frac{2 (- 3 m - 2 n) (- 1 (- 9 m + 8 n))}{- 9 m + 8 n}$

Step 2.6

Cancel the common factor.

$\frac{2 (- 3 m - 2 n) (- 1 (- 9 m + 8 n))}{- 9 m + 8 n}$

Step 2.7

Divide $2 (- 3 m - 2 n) \cdot (- 1)$ by $1$ .

$2 (- 3 m - 2 n) \cdot (- 1)$

Step 3

Apply the distributive property.

$(2 (- 3 m) + 2 (- 2 n)) \cdot - 1$

Step 4

Multiply $- 3$ by $2$ .

$(- 6 m + 2 (- 2 n)) \cdot - 1$

Step 5

Multiply $- 2$ by $2$ .

$(- 6 m - 4 n) \cdot - 1$

Step 6

Apply the distributive property.

$- 6 m \cdot - 1 - 4 n \cdot - 1$

Step 7

Multiply $- 1$ by $- 6$ .

$6 m - 4 n \cdot - 1$

Step 8

Multiply $- 1$ by $- 4$ .

$6 m + 4 n$