Algebra Examples

$g x^{2} - 3 h x^{2} - 6 f y^{2} - g y^{2} + 6 f x^{2} + 3 h y^{2}$

Step 1

Regroup terms.

$- 3 h x^{2} + 3 h y^{2} + g x^{2} - 6 f y^{2} - g y^{2} + 6 f x^{2}$

Step 2

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

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Step 2.1

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

$- 3 h (x^{2}) + 3 h y^{2} + g x^{2} - 6 f y^{2} - g y^{2} + 6 f x^{2}$

Step 2.2

Factor $- 3 h$ out of $3 h y^{2}$ .

$- 3 h (x^{2}) - 3 h (- 1 y^{2}) + g x^{2} - 6 f y^{2} - g y^{2} + 6 f x^{2}$

Step 2.3

Factor $- 3 h$ out of $- 3 h (x^{2}) - 3 h (- 1 y^{2})$ .

$- 3 h (x^{2} - 1 y^{2}) + g x^{2} - 6 f y^{2} - g y^{2} + 6 f x^{2}$

Step 3

Factor.

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Step 3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

$- 3 h ((x + y) (x - y)) + g x^{2} - 6 f y^{2} - g y^{2} + 6 f x^{2}$

Step 3.2

Remove unnecessary parentheses.

$- 3 h (x + y) (x - y) + g x^{2} - 6 f y^{2} - g y^{2} + 6 f x^{2}$

Step 4

Regroup terms.

$- 3 h (x + y) (x - y) + g x^{2} - g y^{2} - 6 f y^{2} + 6 f x^{2}$

Step 5

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

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Step 5.1

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

$- 3 h (x + y) (x - y) + g (x^{2}) - g y^{2} - 6 f y^{2} + 6 f x^{2}$

Step 5.2

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

$- 3 h (x + y) (x - y) + g (x^{2}) + g (- 1 y^{2}) - 6 f y^{2} + 6 f x^{2}$

Step 5.3

Factor $g$ out of $g (x^{2}) + g (- 1 y^{2})$ .

$- 3 h (x + y) (x - y) + g (x^{2} - 1 y^{2}) - 6 f y^{2} + 6 f x^{2}$

Step 6

Factor.

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Step 6.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

$- 3 h (x + y) (x - y) + g ((x + y) (x - y)) - 6 f y^{2} + 6 f x^{2}$

Step 6.2

Remove unnecessary parentheses.

$- 3 h (x + y) (x - y) + g (x + y) (x - y) - 6 f y^{2} + 6 f x^{2}$

Step 7

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

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Step 7.1

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

$- 3 h (x + y) (x - y) + g (x + y) (x - y) + 6 f (- 1 y^{2}) + 6 f x^{2}$

Step 7.2

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

$- 3 h (x + y) (x - y) + g (x + y) (x - y) + 6 f (- 1 y^{2}) + 6 f (x^{2})$

Step 7.3

Factor $6 f$ out of $6 f (- 1 y^{2}) + 6 f (x^{2})$ .

$- 3 h (x + y) (x - y) + g (x + y) (x - y) + 6 f (- 1 y^{2} + x^{2})$

Step 8

Reorder $- 1 y^{2}$ and $x^{2}$ .

$- 3 h (x + y) (x - y) + g (x + y) (x - y) + 6 f (x^{2} - 1 y^{2})$

Step 9

Factor.

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Step 9.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

$- 3 h (x + y) (x - y) + g (x + y) (x - y) + 6 f ((x + y) (x - y))$

Step 9.2

Remove unnecessary parentheses.

$- 3 h (x + y) (x - y) + g (x + y) (x - y) + 6 f (x + y) (x - y)$

Step 10

Factor $(x + y) (x - y)$ out of $g (x + y) (x - y) + 6 f (x + y) (x - y)$ .

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Step 10.1

Factor $(x + y) (x - y)$ out of $g (x + y) (x - y)$ .

$- 3 h (x + y) (x - y) + (x + y) (x - y) (g) + 6 f (x + y) (x - y)$

Step 10.2

Factor $(x + y) (x - y)$ out of $6 f (x + y) (x - y)$ .

$- 3 h (x + y) (x - y) + (x + y) (x - y) (g) + (x + y) (x - y) (6 f)$

Step 10.3

Factor $(x + y) (x - y)$ out of $(x + y) (x - y) (g) + (x + y) (x - y) (6 f)$ .

$- 3 h (x + y) (x - y) + (x + y) (x - y) (g + 6 f)$

Step 11

Factor $(x + y) (x - y)$ out of $- 3 h (x + y) (x - y) + (x + y) (x - y) (g + 6 f)$ .

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Step 11.1

Factor $(x + y) (x - y)$ out of $- 3 h (x + y) (x - y)$ .

$(x + y) (x - y) (- 3 h) + (x + y) (x - y) (g + 6 f)$

Step 11.2

Factor $(x + y) (x - y)$ out of $(x + y) (x - y) (- 3 h) + (x + y) (x - y) (g + 6 f)$ .

$(x + y) (x - y) (- 3 h + g + 6 f)$

Step 12

Factor.

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Step 12.1

Factor $- 1$ out of $- 3 h + g + 6 f$ .

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Step 12.1.1

Reorder the expression.

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Step 12.1.1.1

Move $- 3 h$ .

$(x + y) (x - y) (g + 6 f - 3 h)$

Step 12.1.1.2

Reorder $g$ and $6 f$ .

$(x + y) (x - y) (6 f + g - 3 h)$

Step 12.1.2

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

$(x + y) (x - y) (- (- 6 f) + g - 3 h)$

Step 12.1.3

Factor $- 1$ out of $g$ .

$(x + y) (x - y) (- (- 6 f) - 1 (- g) - 3 h)$

Step 12.1.4

Factor $- 1$ out of $- 3 h$ .

$(x + y) (x - y) (- (- 6 f) - 1 (- g) - (3 h))$

Step 12.1.5

Factor $- 1$ out of $- (- 6 f) - 1 (- g)$ .

$(x + y) (x - y) (- (- 6 f - g) - (3 h))$

Step 12.1.6

Factor $- 1$ out of $- (- 6 f - g) - (3 h)$ .

$(x + y) (x - y) (- (- 6 f - g + 3 h))$

Step 12.2

Remove unnecessary parentheses.

$(x + y) (x - y) \cdot - 1 (- 6 f - g + 3 h)$

Step 13

Factor out negative.

$- ((x + y) (x - y) (- 6 f - g + 3 h))$

Step 14

Remove unnecessary parentheses.

$- (x + y) (x - y) (- 6 f - g + 3 h)$

Step 15

Factor.

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Step 15.1

Factor $- 1$ out of $- 6 f - g + 3 h$ .

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Step 15.1.1

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

$- (x + y) (x - y) (- (6 f) - g + 3 h)$

Step 15.1.2

Factor $- 1$ out of $- g$ .

$- (x + y) (x - y) (- (6 f) - (g) + 3 h)$

Step 15.1.3

Factor $- 1$ out of $3 h$ .

$- (x + y) (x - y) (- (6 f) - (g) - (- 3 h))$

Step 15.1.4

Factor $- 1$ out of $- (6 f) - (g)$ .

$- (x + y) (x - y) (- (6 f + g) - (- 3 h))$

Step 15.1.5

Factor $- 1$ out of $- (6 f + g) - (- 3 h)$ .

$- (x + y) (x - y) (- (6 f + g - 3 h))$

Step 15.2

Remove unnecessary parentheses.

$- (x + y) (x - y) \cdot - 1 (6 f + g - 3 h)$

Step 16

Combine exponents.

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Step 16.1

Factor out negative.

$- (((x + y) (x - y)) \cdot - 1 (6 f + g - 3 h))$

Step 16.2

Multiply $- 1$ by $- 1$ .

$1 (((x + y) (x - y)) (6 f + g - 3 h))$

Step 16.3

Multiply $x + y$ by $1$ .

$(x + y) ((x - y) (6 f + g - 3 h))$

Step 17

Remove unnecessary parentheses.

$(x + y) (x - y) (6 f + g - 3 h)$