Calculus Examples

$x^{2} + 10 x + 25 - 9 y^{4}$

Step 1

Factor using the perfect square rule.

Tap for more steps...

Step 1.1

Rewrite $25$ as $5^{2}$ .

$x^{2} + 10 x + 5^{2} - 9 y^{4}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 1.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 5 + 5^{2} - 9 y^{4}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

${(x + 5)}^{2} - 9 y^{4}$

Step 2

Rewrite $9 y^{4}$ as ${(3 y^{2})}^{2}$ .

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + 5$ and $b = 3 y^{2}$ .

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

Step 4

Multiply $3$ by $- 1$ .

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

Step 5

Factor.

Tap for more steps...

Step 5.1

Factor $- 1$ out of $x + 5 - 3 y^{2}$ .

Tap for more steps...

Step 5.1.1

Reorder the expression.

Tap for more steps...

Step 5.1.1.1

Move $5$ .

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

Step 5.1.1.2

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

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

Step 5.1.2

Factor $- 1$ out of $- 3 y^{2}$ .

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

Step 5.1.3

Factor $- 1$ out of $x$ .

$(x + 5 + 3 y^{2}) (- (3 y^{2}) - 1 (- x) + 5)$

Step 5.1.4

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

$(x + 5 + 3 y^{2}) (- (3 y^{2}) - 1 (- x) - 1 (- 5))$

Step 5.1.5

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

$(x + 5 + 3 y^{2}) (- (3 y^{2} - x) - 1 (- 5))$

Step 5.1.6

Factor $- 1$ out of $- (3 y^{2} - x) - 1 (- 5)$ .

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

Step 5.2

Remove unnecessary parentheses.

$(x + 5 + 3 y^{2}) \cdot - 1 (3 y^{2} - x - 5)$

Step 6

Factor out negative.

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

Step 7

Remove unnecessary parentheses.

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