Algebra Examples

$\frac{5 x^{2} + 14 x - 3}{3 x - 15 x^{2}}$

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 $a \cdot c = 5 \cdot - 3 = - 15$ and whose sum is $b = 14$ .

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Step 1.1.1

Factor $14$ out of $14 x$ .

$\frac{5 x^{2} + 14 (x) - 3}{3 x - 15 x^{2}}$

Step 1.1.2

Rewrite $14$ as $- 1$ plus $15$

$\frac{5 x^{2} + (- 1 + 15) x - 3}{3 x - 15 x^{2}}$

Step 1.1.3

Apply the distributive property.

$\frac{5 x^{2} - 1 x + 15 x - 3}{3 x - 15 x^{2}}$

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.

$\frac{(5 x^{2} - 1 x) + 15 x - 3}{3 x - 15 x^{2}}$

Step 1.2.2

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

$\frac{x (5 x - 1) + 3 (5 x - 1)}{3 x - 15 x^{2}}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 1$ .

$\frac{(5 x - 1) (x + 3)}{3 x - 15 x^{2}}$

Step 2

Factor $3 x$ out of $3 x - 15 x^{2}$ .

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Step 2.1

Factor $3 x$ out of $3 x$ .

$\frac{(5 x - 1) (x + 3)}{3 x (1) - 15 x^{2}}$

Step 2.2

Factor $3 x$ out of $- 15 x^{2}$ .

$\frac{(5 x - 1) (x + 3)}{3 x (1) + 3 x (- 5 x)}$

Step 2.3

Factor $3 x$ out of $3 x (1) + 3 x (- 5 x)$ .

$\frac{(5 x - 1) (x + 3)}{3 x (1 - 5 x)}$

Step 3

Cancel the common factor of $5 x - 1$ and $1 - 5 x$ .

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Step 3.1

Factor $- 1$ out of $5 x$ .

$\frac{(- (- 5 x) - 1) (x + 3)}{3 x (1 - 5 x)}$

Step 3.2

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{(- (- 5 x) - 1 (1)) (x + 3)}{3 x (1 - 5 x)}$

Step 3.3

Factor $- 1$ out of $- (- 5 x) - 1 (1)$ .

$\frac{- (- 5 x + 1) (x + 3)}{3 x (1 - 5 x)}$

Step 3.4

Rewrite $- (- 5 x + 1)$ as $- 1 (- 5 x + 1)$ .

$\frac{- 1 (- 5 x + 1) (x + 3)}{3 x (1 - 5 x)}$

Step 3.5

Reorder terms.

$\frac{- 1 (- 5 x + 1) (x + 3)}{3 x (- 5 x + 1)}$

Step 3.6

Cancel the common factor.

$\frac{- 1 (- 5 x + 1) (x + 3)}{3 x (- 5 x + 1)}$

Step 3.7

Rewrite the expression.

$\frac{- 1 (x + 3)}{3 x}$

Step 4

Move the negative in front of the fraction.

$- \frac{x + 3}{3 x}$