Finite Math Examples

$\frac{2 x}{1 x} + \frac{x + 3}{x^{2} - 1} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{2 x}{1 x} + \frac{x + 3}{x^{2} - 1} - 1 = 0$

Step 2

Simplify $\frac{2 x}{1 x} + \frac{x + 3}{x^{2} - 1} - 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $x$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{2 x}{1 x} + \frac{x + 3}{x^{2} - 1} - 1 = 0$

Step 2.1.1.2

Divide $2$ by $1$ .

$2 + \frac{x + 3}{x^{2} - 1} - 1 = 0$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

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

$2 + \frac{x + 3}{x^{2} - 1^{2}} - 1 = 0$

Step 2.1.2.2

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

$2 + \frac{x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

$2 \cdot \frac{(x + 1) (x - 1)}{(x + 1) (x - 1)} + \frac{x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.3

Combine $2$ and $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

$\frac{2 ((x + 1) (x - 1))}{(x + 1) (x - 1)} + \frac{x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{2 ((x + 1) (x - 1)) + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot 1) (x - 1) + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.2

Multiply $2$ by $1$ .

$\frac{(2 x + 2) (x - 1) + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.3

Expand $(2 x + 2) (x - 1)$ using the FOIL Method.

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Step 2.5.3.1

Apply the distributive property.

$\frac{2 x (x - 1) + 2 (x - 1) + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.3.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 1 + 2 (x - 1) + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.3.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 1 + 2 x + 2 \cdot - 1 + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.4

Simplify and combine like terms.

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Step 2.5.4.1

Simplify each term.

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Step 2.5.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.4.1.1.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot - 1 + 2 x + 2 \cdot - 1 + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 1 + 2 x + 2 \cdot - 1 + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.4.1.2

Multiply $- 1$ by $2$ .

$\frac{2 x^{2} - 2 x + 2 x + 2 \cdot - 1 + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.4.1.3

Multiply $2$ by $- 1$ .

$\frac{2 x^{2} - 2 x + 2 x - 2 + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.4.2

Add $- 2 x$ and $2 x$ .

$\frac{2 x^{2} + 0 - 2 + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.4.3

Add $2 x^{2}$ and $0$ .

$\frac{2 x^{2} - 2 + x + 3}{(x + 1) (x - 1)} - 1 = 0$

Step 2.5.5

Add $- 2$ and $3$ .

$\frac{2 x^{2} + x + 1}{(x + 1) (x - 1)} - 1 = 0$

Step 2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

$\frac{2 x^{2} + x + 1}{(x + 1) (x - 1)} - 1 \cdot \frac{(x + 1) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.7

Combine $- 1$ and $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

$\frac{2 x^{2} + x + 1}{(x + 1) (x - 1)} + \frac{- ((x + 1) (x - 1))}{(x + 1) (x - 1)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{2 x^{2} + x + 1 - ((x + 1) (x - 1))}{(x + 1) (x - 1)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{2 x^{2} + x + 1 + (- x - 1 \cdot 1) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.2

Multiply $- 1$ by $1$ .

$\frac{2 x^{2} + x + 1 + (- x - 1) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.3

Expand $(- x - 1) (x - 1)$ using the FOIL Method.

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Step 2.9.3.1

Apply the distributive property.

$\frac{2 x^{2} + x + 1 - x (x - 1) - 1 (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.3.2

Apply the distributive property.

$\frac{2 x^{2} + x + 1 - x \cdot x - x \cdot - 1 - 1 (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.9.3.3

Apply the distributive property.

$\frac{2 x^{2} + x + 1 - x \cdot x - x \cdot - 1 - 1 x - 1 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4

Simplify and combine like terms.

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Step 2.9.4.1

Simplify each term.

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Step 2.9.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.9.4.1.1.1

Move $x$ .

$\frac{2 x^{2} + x + 1 - (x \cdot x) - x \cdot - 1 - 1 x - 1 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x + 1 - x^{2} - x \cdot - 1 - 1 x - 1 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4.1.2

Multiply $- x \cdot - 1$ .

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Step 2.9.4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} + x + 1 - x^{2} + 1 x - 1 x - 1 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4.1.2.2

Multiply $x$ by $1$ .

$\frac{2 x^{2} + x + 1 - x^{2} + x - 1 x - 1 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{2} + x + 1 - x^{2} + x - x - 1 \cdot - 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4.1.4

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} + x + 1 - x^{2} + x - x + 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4.2

Subtract $x$ from $x$ .

$\frac{2 x^{2} + x + 1 - x^{2} + 0 + 1}{(x + 1) (x - 1)} = 0$

Step 2.9.4.3

Add $- x^{2}$ and $0$ .

$\frac{2 x^{2} + x + 1 - x^{2} + 1}{(x + 1) (x - 1)} = 0$

Step 2.9.5

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} + x + 1 + 1}{(x + 1) (x - 1)} = 0$

Step 2.9.6

Add $1$ and $1$ .

$\frac{x^{2} + x + 2}{(x + 1) (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} + x + 2 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 4.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2}}{2 \cdot 1}$

Step 4.3.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Step 4.3.1.3

Subtract $8$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 4.3.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 4.3.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 4.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2}$

Step 4.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Step 4.4.1.3

Subtract $8$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 4.4.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 4.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2}$

Step 4.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{7}}{2}$

Step 4.4.4

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

$x = \frac{- 1 \cdot 1 + i \sqrt{7}}{2}$

Step 4.4.5

Factor $- 1$ out of $i \sqrt{7}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{7})}{2}$

Step 4.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{7})$ .

$x = \frac{- 1 (1 - i \sqrt{7})}{2}$

Step 4.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{7}}{2}$

Step 4.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8}}{2 \cdot 1}$

Step 4.5.1.3

Subtract $8$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 4.5.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 4.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2}$

Step 4.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{7}}{2}$

Step 4.5.4

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

$x = \frac{- 1 \cdot 1 - i \sqrt{7}}{2}$

Step 4.5.5

Factor $- 1$ out of $- i \sqrt{7}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{7})}{2}$

Step 4.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{7})$ .

$x = \frac{- 1 (1 + i \sqrt{7})}{2}$

Step 4.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{7}}{2}$

Step 4.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{7}}{2}, - \frac{1 + i \sqrt{7}}{2}$