Basic Math Examples

$\frac{25 x^{2} + 10 x + 1}{1 - 25 x^{2}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$\frac{{(5 x)}^{2} + 10 x + 1}{1 - 25 x^{2}}$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(5 x)}^{2} + 10 x + 1^{2}}{1 - 25 x^{2}}$

Step 1.3

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

$10 x = 2 \cdot (5 x) \cdot 1$

Step 1.4

Rewrite the polynomial.

$\frac{{(5 x)}^{2} + 2 \cdot (5 x) \cdot 1 + 1^{2}}{1 - 25 x^{2}}$

Step 1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 5 x$ and $b = 1$ .

$\frac{{(5 x + 1)}^{2}}{1 - 25 x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\frac{{(5 x + 1)}^{2}}{1^{2} - 25 x^{2}}$

Step 2.2

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$\frac{{(5 x + 1)}^{2}}{1^{2} - {(5 x)}^{2}}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 5 x$ .

$\frac{{(5 x + 1)}^{2}}{(1 + 5 x) (1 - (5 x))}$

Step 2.4

Multiply $5$ by $- 1$ .

$\frac{{(5 x + 1)}^{2}}{(1 + 5 x) (1 - 5 x)}$

Step 3

Cancel the common factor of ${(5 x + 1)}^{2}$ and $1 + 5 x$ .

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Step 3.1

Reorder terms.

$\frac{{(5 x + 1)}^{2}}{(5 x + 1) (1 - 5 x)}$

Step 3.2

Factor $5 x + 1$ out of ${(5 x + 1)}^{2}$ .

$\frac{(5 x + 1) (5 x + 1)}{(5 x + 1) (1 - 5 x)}$

Step 3.3

Cancel the common factors.

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Step 3.3.1

Cancel the common factor.

$\frac{(5 x + 1) (5 x + 1)}{(5 x + 1) (1 - 5 x)}$

Step 3.3.2

Rewrite the expression.

$\frac{5 x + 1}{1 - 5 x}$