Algebra Examples

$3 {(2 x - y)}^{2} + 2 y (2 x - y) - 5 y^{2}$

Step 1

Add parentheses.

$3 {(2 x - y)}^{2} + 2 (y (2 x - y)) - 5 y^{2}$

Step 2

Let $u = y (2 x - y)$ . Substitute $u$ for all occurrences of $y (2 x - y)$ .

$12 x^{2} - 12 x y - 2 y^{2} + 2 u$

Step 3

Factor $2$ out of $12 x^{2} - 12 x y - 2 y^{2} + 2 u$ .

Tap for more steps...

Step 3.1

Factor $2$ out of $12 x^{2}$ .

$2 (6 x^{2}) - 12 x y - 2 y^{2} + 2 u$

Step 3.2

Factor $2$ out of $- 12 x y$ .

$2 (6 x^{2}) + 2 (- 6 x y) - 2 y^{2} + 2 u$

Step 3.3

Factor $2$ out of $- 2 y^{2}$ .

$2 (6 x^{2}) + 2 (- 6 x y) + 2 (- y^{2}) + 2 u$

Step 3.4

Factor $2$ out of $2 u$ .

$2 (6 x^{2}) + 2 (- 6 x y) + 2 (- y^{2}) + 2 (u)$

Step 3.5

Factor $2$ out of $2 (6 x^{2}) + 2 (- 6 x y)$ .

$2 (6 x^{2} - 6 x y) + 2 (- y^{2}) + 2 (u)$

Step 3.6

Factor $2$ out of $2 (6 x^{2} - 6 x y) + 2 (- y^{2})$ .

$2 (6 x^{2} - 6 x y - y^{2}) + 2 (u)$

Step 3.7

Factor $2$ out of $2 (6 x^{2} - 6 x y - y^{2}) + 2 (u)$ .

$2 (6 x^{2} - 6 x y - y^{2} + u)$

Step 4

Replace all occurrences of $u$ with $y (2 x - y)$ .

$2 (6 x^{2} - 6 x y - y^{2} + y (2 x - y))$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

Apply the distributive property.

$2 (6 x^{2} - 6 x y - y^{2} + y (2 x) + y (- y))$

Step 5.1.2

Rewrite using the commutative property of multiplication.

$2 (6 x^{2} - 6 x y - y^{2} + 2 y x + y (- y))$

Step 5.1.3

Rewrite using the commutative property of multiplication.

$2 (6 x^{2} - 6 x y - y^{2} + 2 y x - y \cdot y)$

Step 5.1.4

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 5.1.4.1

Move $y$ .

$2 (6 x^{2} - 6 x y - y^{2} + 2 y x - (y \cdot y))$

Step 5.1.4.2

Multiply $y$ by $y$ .

$2 (6 x^{2} - 6 x y - y^{2} + 2 y x - y^{2})$

Step 5.2

Add $- 6 x y$ and $2 y x$ .

Tap for more steps...

Step 5.2.1

Move $y$ .

$2 (6 x^{2} - y^{2} - 6 x y + 2 x y - y^{2})$

Step 5.2.2

Add $- 6 x y$ and $2 x y$ .

$2 (6 x^{2} - y^{2} - 4 x y - y^{2})$

Step 5.3

Subtract $y^{2}$ from $- y^{2}$ .

$2 (6 x^{2} - 2 y^{2} - 4 x y)$

Step 6

Factor $2$ out of $6 x^{2} - 2 y^{2} - 4 x y$ .

Tap for more steps...

Step 6.1

Factor $2$ out of $6 x^{2}$ .

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

Step 6.2

Factor $2$ out of $- 2 y^{2}$ .

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

Step 6.3

Factor $2$ out of $- 4 x y$ .

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

Step 6.4

Factor $2$ out of $2 (3 x^{2}) + 2 (- y^{2})$ .

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

Step 6.5

Factor $2$ out of $2 (3 x^{2} - y^{2}) + 2 (- 2 x y)$ .

$2 (2 (3 x^{2} - y^{2} - 2 x y))$

Step 7

Factor.

Tap for more steps...

Step 7.1

Factor.

Tap for more steps...

Step 7.1.1

Factor by grouping.

Tap for more steps...

Step 7.1.1.1

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

Tap for more steps...

Step 7.1.1.1.1

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

$2 (2 (3 x^{2} - 2 x y - y^{2}))$

Step 7.1.1.1.2

Factor $- 2$ out of $- 2 x y$ .

$2 (2 (3 x^{2} - 2 (x y) - y^{2}))$

Step 7.1.1.1.3

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

$2 (2 (3 x^{2} + (1 - 3) (x y) - y^{2}))$

Step 7.1.1.1.4

Apply the distributive property.

$2 (2 (3 x^{2} + 1 (x y) - 3 (x y) - y^{2}))$

Step 7.1.1.1.5

Multiply $x y$ by $1$ .

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

Step 7.1.1.1.6

Remove unnecessary parentheses.

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

Step 7.1.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 7.1.1.2.1

Group the first two terms and the last two terms.

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

Step 7.1.1.2.2

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

$2 (2 (x (3 x + y) - y (3 x + y)))$

Step 7.1.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + y$ .

$2 (2 ((3 x + y) (x - y)))$

Step 7.1.2

Remove unnecessary parentheses.

$2 (2 (3 x + y) (x - y))$

Step 7.2

Remove unnecessary parentheses.

$2 \cdot 2 (3 x + y) (x - y)$

Step 8

Multiply $2$ by $2$ .

$4 (3 x + y) (x - y)$