Algebra Examples

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

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

Step 1

Factor by grouping.

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Step 1.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = - 63 \cdot 30 = - 1890

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

b = 19

$b = 19$ .

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Step 1.1.1

Reorder terms.

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

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

Step 1.1.2

Reorder

30 c^{2}

$30 c^{2}$ and

19 a c

$19 a c$ .

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

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

Step 1.1.3

Factor

19

$19$ out of

19 a c

$19 a c$ .

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

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

Step 1.1.4

Rewrite

19

$19$ as

- 35

$- 35$ plus

54

$54$

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

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

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}

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

Step 1.1.6

Move parentheses.

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

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

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

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

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}

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

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}

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

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

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

Step 1.3

Factor the polynomial by factoring out the greatest common factor,

- 9 a - 5 c

$- 9 a - 5 c$ .

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

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

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

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

Step 2

Cancel the common factor of

7 a - 6 c

$7 a - 6 c$ and

6 c - 7 a

$6 c - 7 a$ .

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Step 2.1

Factor

- 1

$- 1$ out of

7 a

$7 a$ .

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

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

Step 2.2

Factor

- 1

$- 1$ out of

- 6 c

$- 6 c$ .

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

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

Step 2.3

Factor

- 1

$- 1$ out of

- (- 7 a) - (6 c)

$- (- 7 a) - (6 c)$ .

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

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

Step 2.4

Rewrite

- (- 7 a + 6 c)

$- (- 7 a + 6 c)$ as

- 1 (- 7 a + 6 c)

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

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

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

Step 2.5

Reorder terms.

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

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

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

$- 5$ .

$9 a + 5 c$