Finite Math Examples

$\frac{2 x}{x + 2} + \frac{x + 2}{2 x} = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\frac{2 x}{x + 2} + \frac{x + 2}{2 x} - 2 = 0$

Step 2

Simplify $\frac{2 x}{x + 2} + \frac{x + 2}{2 x} - 2$ .

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Step 2.1

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

$\frac{2 x}{x + 2} \cdot \frac{2 x}{2 x} + \frac{x + 2}{2 x} - 2 = 0$

Step 2.2

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

$\frac{2 x}{x + 2} \cdot \frac{2 x}{2 x} + \frac{x + 2}{2 x} \cdot \frac{x + 2}{x + 2} - 2 = 0$

Step 2.3

Write each expression with a common denominator of $(x + 2) \cdot 2 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{2 x}{x + 2}$ by $\frac{2 x}{2 x}$ .

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

Step 2.3.2

Multiply $\frac{x + 2}{2 x}$ by $\frac{x + 2}{x + 2}$ .

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

Step 2.3.3

Reorder the factors of $(x + 2) (2 x)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Rewrite using the commutative property of multiplication.

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

Step 2.5.2

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

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Step 2.5.2.1

Move $x$ .

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

Step 2.5.2.2

Multiply $x$ by $x$ .

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

Step 2.5.3

Multiply $2$ by $2$ .

$\frac{4 x^{2} + (x + 2) (x + 2)}{2 x (x + 2)} - 2 = 0$

Step 2.5.4

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

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Step 2.5.4.1

Apply the distributive property.

$\frac{4 x^{2} + x (x + 2) + 2 (x + 2)}{2 x (x + 2)} - 2 = 0$

Step 2.5.4.2

Apply the distributive property.

$\frac{4 x^{2} + x \cdot x + x \cdot 2 + 2 (x + 2)}{2 x (x + 2)} - 2 = 0$

Step 2.5.4.3

Apply the distributive property.

$\frac{4 x^{2} + x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2}{2 x (x + 2)} - 2 = 0$

Step 2.5.5

Simplify and combine like terms.

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Step 2.5.5.1

Simplify each term.

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Step 2.5.5.1.1

Multiply $x$ by $x$ .

$\frac{4 x^{2} + x^{2} + x \cdot 2 + 2 x + 2 \cdot 2}{2 x (x + 2)} - 2 = 0$

Step 2.5.5.1.2

Move $2$ to the left of $x$ .

$\frac{4 x^{2} + x^{2} + 2 \cdot x + 2 x + 2 \cdot 2}{2 x (x + 2)} - 2 = 0$

Step 2.5.5.1.3

Multiply $2$ by $2$ .

$\frac{4 x^{2} + x^{2} + 2 x + 2 x + 4}{2 x (x + 2)} - 2 = 0$

Step 2.5.5.2

Add $2 x$ and $2 x$ .

$\frac{4 x^{2} + x^{2} + 4 x + 4}{2 x (x + 2)} - 2 = 0$

Step 2.5.6

Add $4 x^{2}$ and $x^{2}$ .

$\frac{5 x^{2} + 4 x + 4}{2 x (x + 2)} - 2 = 0$

Step 2.6

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

$\frac{5 x^{2} + 4 x + 4}{2 x (x + 2)} - 2 \cdot \frac{2 x (x + 2)}{2 x (x + 2)} = 0$

Step 2.7

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

$\frac{5 x^{2} + 4 x + 4}{2 x (x + 2)} + \frac{- 2 (2 x (x + 2))}{2 x (x + 2)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{5 x^{2} + 4 x + 4 - 2 (2 x (x + 2))}{2 x (x + 2)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 2$ by $2$ .

$\frac{5 x^{2} + 4 x + 4 - 4 x (x + 2)}{2 x (x + 2)} = 0$

Step 2.9.2

Apply the distributive property.

$\frac{5 x^{2} + 4 x + 4 - 4 x \cdot x - 4 x \cdot 2}{2 x (x + 2)} = 0$

Step 2.9.3

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

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Step 2.9.3.1

Move $x$ .

$\frac{5 x^{2} + 4 x + 4 - 4 (x \cdot x) - 4 x \cdot 2}{2 x (x + 2)} = 0$

Step 2.9.3.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 4 x + 4 - 4 x^{2} - 4 x \cdot 2}{2 x (x + 2)} = 0$

Step 2.9.4

Multiply $2$ by $- 4$ .

$\frac{5 x^{2} + 4 x + 4 - 4 x^{2} - 8 x}{2 x (x + 2)} = 0$

Step 2.9.5

Subtract $4 x^{2}$ from $5 x^{2}$ .

$\frac{x^{2} + 4 x + 4 - 8 x}{2 x (x + 2)} = 0$

Step 2.9.6

Subtract $8 x$ from $4 x$ .

$\frac{x^{2} - 4 x + 4}{2 x (x + 2)} = 0$

Step 2.9.7

Factor using the perfect square rule.

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Step 2.9.7.1

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 4 x + 2^{2}}{2 x (x + 2)} = 0$

Step 2.9.7.2

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

$4 x = 2 \cdot x \cdot 2$

Step 2.9.7.3

Rewrite the polynomial.

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

Step 2.9.7.4

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

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

Step 3

Set the numerator equal to zero.

${(x - 2)}^{2} = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$