Algebra Examples

$\frac{6 c^{2} + 13 c - 63}{6 c^{2} - 17 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 1

Factor by grouping.

Tap for more steps...

Step 1.1

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 $a \cdot c = 6 \cdot - 63 = - 378$ and whose sum is $b = 13$ .

Tap for more steps...

Step 1.1.1

Factor $13$ out of $13 c$ .

$\frac{6 c^{2} + 13 (c) - 63}{6 c^{2} - 17 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 1.1.2

Rewrite $13$ as $- 14$ plus $27$

$\frac{6 c^{2} + (- 14 + 27) c - 63}{6 c^{2} - 17 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 1.1.3

Apply the distributive property.

$\frac{6 c^{2} - 14 c + 27 c - 63}{6 c^{2} - 17 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(6 c^{2} - 14 c) + 27 c - 63}{6 c^{2} - 17 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 1.2.2

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

$\frac{2 c (3 c - 7) + 9 (3 c - 7)}{6 c^{2} - 17 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 c - 7$ .

$\frac{(3 c - 7) (2 c + 9)}{6 c^{2} - 17 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 2

Factor by grouping.

Tap for more steps...

Step 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 $a \cdot c = 6 \cdot 7 = 42$ and whose sum is $b = - 17$ .

Tap for more steps...

Step 2.1.1

Factor $- 17$ out of $- 17 c$ .

$\frac{(3 c - 7) (2 c + 9)}{6 c^{2} - 17 (c) + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 2.1.2

Rewrite $- 17$ as $- 3$ plus $- 14$

$\frac{(3 c - 7) (2 c + 9)}{6 c^{2} + (- 3 - 14) c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 2.1.3

Apply the distributive property.

$\frac{(3 c - 7) (2 c + 9)}{6 c^{2} - 3 c - 14 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(3 c - 7) (2 c + 9)}{(6 c^{2} - 3 c) - 14 c + 7} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 2.2.2

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

$\frac{(3 c - 7) (2 c + 9)}{3 c (2 c - 1) - 7 (2 c - 1)} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 c - 1$ .

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{2 c^{2} - 9 c + 4}{12 c + 54}$

Step 3

Factor by grouping.

Tap for more steps...

Step 3.1

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 $a \cdot c = 2 \cdot 4 = 8$ and whose sum is $b = - 9$ .

Tap for more steps...

Step 3.1.1

Factor $- 9$ out of $- 9 c$ .

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{2 c^{2} - 9 (c) + 4}{12 c + 54}$

Step 3.1.2

Rewrite $- 9$ as $- 1$ plus $- 8$

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{2 c^{2} + (- 1 - 8) c + 4}{12 c + 54}$

Step 3.1.3

Apply the distributive property.

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{2 c^{2} - 1 c - 8 c + 4}{12 c + 54}$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c^{2} - 1 c) - 8 c + 4}{12 c + 54}$

Step 3.2.2

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

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{c (2 c - 1) - 4 (2 c - 1)}{12 c + 54}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 c - 1$ .

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{12 c + 54}$

Step 4

Factor $6$ out of $12 c + 54$ .

Tap for more steps...

Step 4.1

Factor $6$ out of $12 c$ .

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{6 (2 c) + 54}$

Step 4.2

Factor $6$ out of $54$ .

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{6 (2 c) + 6 (9)}$

Step 4.3

Factor $6$ out of $6 (2 c) + 6 (9)$ .

$\frac{(3 c - 7) (2 c + 9)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{6 (2 c + 9)}$

Step 5

Cancel the common factor of $2 c + 9$ .

Tap for more steps...

Step 5.1

Factor $2 c + 9$ out of $(3 c - 7) (2 c + 9)$ .

$\frac{(2 c + 9) (3 c - 7)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{6 (2 c + 9)}$

Step 5.2

Factor $2 c + 9$ out of $6 (2 c + 9)$ .

$\frac{(2 c + 9) (3 c - 7)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{(2 c + 9) \cdot 6}$

Step 5.3

Cancel the common factor.

$\frac{(2 c + 9) (3 c - 7)}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{(2 c + 9) \cdot 6}$

Step 5.4

Rewrite the expression.

$\frac{3 c - 7}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{6}$

Step 6

Cancel the common factor of $2 c - 1$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\frac{3 c - 7}{(2 c - 1) (3 c - 7)} \cdot \frac{(2 c - 1) (c - 4)}{6}$

Step 6.2

Rewrite the expression.

$\frac{3 c - 7}{3 c - 7} \cdot \frac{c - 4}{6}$

Step 7

Multiply $\frac{3 c - 7}{3 c - 7}$ by $\frac{c - 4}{6}$ .

$\frac{(3 c - 7) (c - 4)}{(3 c - 7) \cdot 6}$

Step 8

Cancel the common factor of $3 c - 7$ .

Tap for more steps...

Step 8.1

Cancel the common factor.

$\frac{(3 c - 7) (c - 4)}{(3 c - 7) \cdot 6}$

Step 8.2

Rewrite the expression.

$\frac{c - 4}{6}$