Algebra Examples

$\frac{4 x^{2} - 4 x y - 3 y^{2}}{2 x^{2} - x y - 3 y^{2}}$

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = - 4$ .

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Step 1.1.1

Reorder terms.

$\frac{4 x^{2} - 3 y^{2} - 4 x y}{2 x^{2} - x y - 3 y^{2}}$

Step 1.1.2

Reorder $- 3 y^{2}$ and $- 4 x y$ .

$\frac{4 x^{2} - 4 x y - 3 y^{2}}{2 x^{2} - x y - 3 y^{2}}$

Step 1.1.3

Factor $- 4$ out of $- 4 x y$ .

$\frac{4 x^{2} - 4 (x y) - 3 y^{2}}{2 x^{2} - x y - 3 y^{2}}$

Step 1.1.4

Rewrite $- 4$ as $2$ plus $- 6$

$\frac{4 x^{2} + (2 - 6) (x y) - 3 y^{2}}{2 x^{2} - x y - 3 y^{2}}$

Step 1.1.5

Apply the distributive property.

$\frac{4 x^{2} + 2 (x y) - 6 (x y) - 3 y^{2}}{2 x^{2} - x y - 3 y^{2}}$

Step 1.1.6

Move parentheses.

$\frac{4 x^{2} + 2 x y - 6 x y - 3 y^{2}}{2 x^{2} - x y - 3 y^{2}}$

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.

$\frac{(4 x^{2} + 2 x y) - 6 x y - 3 y^{2}}{2 x^{2} - x y - 3 y^{2}}$

Step 1.2.2

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

$\frac{2 x (2 x + y) - 3 y (2 x + y)}{2 x^{2} - x y - 3 y^{2}}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + y$ .

$\frac{(2 x + y) (2 x - 3 y)}{2 x^{2} - x y - 3 y^{2}}$

Step 2

Factor by grouping.

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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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

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Step 2.1.1

Reorder terms.

$\frac{(2 x + y) (2 x - 3 y)}{2 x^{2} - 3 y^{2} - x y}$

Step 2.1.2

Reorder $- 3 y^{2}$ and $- x y$ .

$\frac{(2 x + y) (2 x - 3 y)}{2 x^{2} - x y - 3 y^{2}}$

Step 2.1.3

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

$\frac{(2 x + y) (2 x - 3 y)}{2 x^{2} - (x y) - 3 y^{2}}$

Step 2.1.4

Rewrite $- 1$ as $2$ plus $- 3$

$\frac{(2 x + y) (2 x - 3 y)}{2 x^{2} + (2 - 3) (x y) - 3 y^{2}}$

Step 2.1.5

Apply the distributive property.

$\frac{(2 x + y) (2 x - 3 y)}{2 x^{2} + 2 (x y) - 3 (x y) - 3 y^{2}}$

Step 2.1.6

Move parentheses.

$\frac{(2 x + y) (2 x - 3 y)}{2 x^{2} + 2 x y - 3 x y - 3 y^{2}}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(2 x + y) (2 x - 3 y)}{(2 x^{2} + 2 x y) - 3 x y - 3 y^{2}}$

Step 2.2.2

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

$\frac{(2 x + y) (2 x - 3 y)}{2 x (x + y) - 3 y (x + y)}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $x + y$ .

$\frac{(2 x + y) (2 x - 3 y)}{(x + y) (2 x - 3 y)}$

Step 3

Cancel the common factor of $2 x - 3 y$ .

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Step 3.1

Cancel the common factor.

$\frac{(2 x + y) (2 x - 3 y)}{(x + y) (2 x - 3 y)}$

Step 3.2

Rewrite the expression.

$\frac{2 x + y}{x + y}$