Basic Math Examples

$81 {(2 x + 3 z)}^{2} - 4 y^{4}$

Step 1

Rewrite $81 {(2 x + 3 z)}^{2}$ as ${(9 (2 x + 3 z))}^{2}$ .

${(9 (2 x + 3 z))}^{2} - 4 y^{4}$

Step 2

Rewrite $4 y^{4}$ as ${(2 y^{2})}^{2}$ .

${(9 (2 x + 3 z))}^{2} - {(2 y^{2})}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9 (2 x + 3 z)$ and $b = 2 y^{2}$ .

$(9 (2 x + 3 z) + 2 y^{2}) (9 (2 x + 3 z) - (2 y^{2}))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(9 (2 x) + 9 (3 z) + 2 y^{2}) (9 (2 x + 3 z) - (2 y^{2}))$

Step 4.2

Multiply $2$ by $9$ .

$(18 x + 9 (3 z) + 2 y^{2}) (9 (2 x + 3 z) - (2 y^{2}))$

Step 4.3

Multiply $3$ by $9$ .

$(18 x + 27 z + 2 y^{2}) (9 (2 x + 3 z) - (2 y^{2}))$

Step 4.4

Apply the distributive property.

$(18 x + 27 z + 2 y^{2}) (9 (2 x) + 9 (3 z) - (2 y^{2}))$

Step 4.5

Multiply $2$ by $9$ .

$(18 x + 27 z + 2 y^{2}) (18 x + 9 (3 z) - (2 y^{2}))$

Step 4.6

Multiply $3$ by $9$ .

$(18 x + 27 z + 2 y^{2}) (18 x + 27 z - (2 y^{2}))$

Step 4.7

Multiply $2$ by $- 1$ .

$(18 x + 27 z + 2 y^{2}) (18 x + 27 z - 2 y^{2})$

Step 5

Factor.

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Step 5.1

Factor $- 1$ out of $18 x + 27 z - 2 y^{2}$ .

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Step 5.1.1

Reorder the expression.

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Step 5.1.1.1

Move $27 z$ .

$(18 x + 27 z + 2 y^{2}) (18 x - 2 y^{2} + 27 z)$

Step 5.1.1.2

Reorder $18 x$ and $- 2 y^{2}$ .

$(18 x + 27 z + 2 y^{2}) (- 2 y^{2} + 18 x + 27 z)$

Step 5.1.2

Factor $- 1$ out of $- 2 y^{2}$ .

$(18 x + 27 z + 2 y^{2}) (- (2 y^{2}) + 18 x + 27 z)$

Step 5.1.3

Factor $- 1$ out of $18 x$ .

$(18 x + 27 z + 2 y^{2}) (- (2 y^{2}) - (- 18 x) + 27 z)$

Step 5.1.4

Factor $- 1$ out of $27 z$ .

$(18 x + 27 z + 2 y^{2}) (- (2 y^{2}) - (- 18 x) - (- 27 z))$

Step 5.1.5

Factor $- 1$ out of $- (2 y^{2}) - (- 18 x)$ .

$(18 x + 27 z + 2 y^{2}) (- (2 y^{2} - 18 x) - (- 27 z))$

Step 5.1.6

Factor $- 1$ out of $- (2 y^{2} - 18 x) - (- 27 z)$ .

$(18 x + 27 z + 2 y^{2}) (- (2 y^{2} - 18 x - 27 z))$

Step 5.2

Remove unnecessary parentheses.

$(18 x + 27 z + 2 y^{2}) \cdot - 1 (2 y^{2} - 18 x - 27 z)$

Step 6

Factor out negative.

$- ((18 x + 27 z + 2 y^{2}) (2 y^{2} - 18 x - 27 z))$

Step 7

Remove unnecessary parentheses.

$- (18 x + 27 z + 2 y^{2}) (2 y^{2} - 18 x - 27 z)$