Algebra Examples

$\frac{30 c^{2} + 19 a c - 63 a^{2}}{6 c - 7 a}$

Step 1

Factor by grouping.

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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 $a \cdot c = - 63 \cdot 30 = - 1890$ and whose sum is $b = 19$ .

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Step 1.1.1

Reorder terms.

$\frac{- 63 a^{2} + 30 c^{2} + 19 a c}{6 c - 7 a}$

Step 1.1.2

Reorder $30 c^{2}$ and $19 a c$ .

$\frac{- 63 a^{2} + 19 a c + 30 c^{2}}{6 c - 7 a}$

Step 1.1.3

Factor $19$ out of $19 a c$ .

$\frac{- 63 a^{2} + 19 (a c) + 30 c^{2}}{6 c - 7 a}$

Step 1.1.4

Rewrite $19$ as $- 35$ plus $54$

$\frac{- 63 a^{2} + (- 35 + 54) (a c) + 30 c^{2}}{6 c - 7 a}$

Step 1.1.5

Apply the distributive property.

$\frac{- 63 a^{2} - 35 (a c) + 54 (a c) + 30 c^{2}}{6 c - 7 a}$

Step 1.1.6

Move parentheses.

$\frac{- 63 a^{2} - 35 a c + 54 a c + 30 c^{2}}{6 c - 7 a}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(- 63 a^{2} - 35 a c) + 54 a c + 30 c^{2}}{6 c - 7 a}$

Step 1.2.2

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

$\frac{7 a (- 9 a - 5 c) - 6 c (- 9 a - 5 c)}{6 c - 7 a}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $- 9 a - 5 c$ .

$\frac{(- 9 a - 5 c) (7 a - 6 c)}{6 c - 7 a}$

Step 2

Cancel the common factor of $7 a - 6 c$ and $6 c - 7 a$ .

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Step 2.1

Factor $- 1$ out of $7 a$ .

$\frac{(- 9 a - 5 c) (- (- 7 a) - 6 c)}{6 c - 7 a}$

Step 2.2

Factor $- 1$ out of $- 6 c$ .

$\frac{(- 9 a - 5 c) (- (- 7 a) - (6 c))}{6 c - 7 a}$

Step 2.3

Factor $- 1$ out of $- (- 7 a) - (6 c)$ .

$\frac{(- 9 a - 5 c) (- (- 7 a + 6 c))}{6 c - 7 a}$

Step 2.4

Rewrite $- (- 7 a + 6 c)$ as $- 1 (- 7 a + 6 c)$ .

$\frac{(- 9 a - 5 c) (- 1 (- 7 a + 6 c))}{6 c - 7 a}$

Step 2.5

Reorder terms.

$\frac{(- 9 a - 5 c) (- 1 (- 7 a + 6 c))}{- 7 a + 6 c}$

Step 2.6

Cancel the common factor.

$\frac{(- 9 a - 5 c) (- 1 (- 7 a + 6 c))}{- 7 a + 6 c}$

Step 2.7

Divide $(- 9 a - 5 c) \cdot (- 1)$ by $1$ .

$(- 9 a - 5 c) \cdot (- 1)$

Step 3

Apply the distributive property.

$- 9 a \cdot - 1 - 5 c \cdot - 1$

Step 4

Multiply.

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Step 4.1

Multiply $- 1$ by $- 9$ .

$9 a - 5 c \cdot - 1$

Step 4.2

Multiply $- 1$ by $- 5$ .

$9 a + 5 c$