Algebra Examples

$\frac{100 x^{2} - 100}{1 - x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $100$ out of $100 x^{2} - 100$ .

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Step 1.1.1

Factor $100$ out of $100 x^{2}$ .

$\frac{100 (x^{2}) - 100}{1 - x^{2}}$

Step 1.1.2

Factor $100$ out of $- 100$ .

$\frac{100 (x^{2}) + 100 (- 1)}{1 - x^{2}}$

Step 1.1.3

Factor $100$ out of $100 (x^{2}) + 100 (- 1)$ .

$\frac{100 (x^{2} - 1)}{1 - x^{2}}$

Step 1.2

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

$\frac{100 (x^{2} - 1^{2})}{1 - x^{2}}$

Step 1.3

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

$\frac{100 (x + 1) (x - 1)}{1 - x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{100 (x + 1) (x - 1)}{1^{2} - x^{2}}$

Step 2.2

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 $b = x$ .

$\frac{100 (x + 1) (x - 1)}{(1 + x) (1 - x)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x + 1$ and $1 + x$ .

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Step 3.1.1

Reorder terms.

$\frac{100 (x + 1) (x - 1)}{(x + 1) (1 - x)}$

Step 3.1.2

Cancel the common factor.

$\frac{100 (x + 1) (x - 1)}{(x + 1) (1 - x)}$

Step 3.1.3

Rewrite the expression.

$\frac{100 (x - 1)}{1 - x}$

Step 3.2

Cancel the common factor of $x - 1$ and $1 - x$ .

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Step 3.2.1

Factor $- 1$ out of $x$ .

$\frac{100 (- 1 (- x) - 1)}{1 - x}$

Step 3.2.2

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

$\frac{100 (- 1 (- x) - 1 (1))}{1 - x}$

Step 3.2.3

Factor $- 1$ out of $- 1 (- x) - 1 (1)$ .

$\frac{100 (- 1 (- x + 1))}{1 - x}$

Step 3.2.4

Reorder terms.

$\frac{100 (- 1 (- x + 1))}{- x + 1}$

Step 3.2.5

Cancel the common factor.

$\frac{100 (- 1 (- x + 1))}{- x + 1}$

Step 3.2.6

Divide $100 \cdot (- 1)$ by $1$ .

$100 \cdot (- 1)$

Step 3.3

Multiply $100$ by $- 1$ .

$- 100$