Finite Math Examples

$20 x - 9 = \frac{20}{x}$

Step 1

Add $9$ to both sides of the equation.

$20 x = \frac{20}{x} + 9$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in $20 x = \frac{20}{x} + 9$ by $x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $20 x = \frac{20}{x} + 9$ by $x$ .

$20 x \cdot x = \frac{20}{x} x + 9 x$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Move $x$ .

$20 (x \cdot x) = \frac{20}{x} x + 9 x$

Step 3.2.1.2

Multiply $x$ by $x$ .

$20 x^{2} = \frac{20}{x} x + 9 x$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

Cancel the common factor.

$20 x^{2} = \frac{20}{x} x + 9 x$

Step 3.3.1.2

Rewrite the expression.

$20 x^{2} = 20 + 9 x$

Step 4

Solve the equation.

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Step 4.1

Subtract $9 x$ from both sides of the equation.

$20 x^{2} - 9 x = 20$

Step 4.2

Subtract $20$ from both sides of the equation.

$20 x^{2} - 9 x - 20 = 0$

Step 4.3

Factor by grouping.

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Step 4.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 20 \cdot - 20 = - 400$ and whose sum is $b = - 9$ .

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Step 4.3.1.1

Factor $- 9$ out of $- 9 x$ .

$20 x^{2} - 9 x - 20 = 0$

Step 4.3.1.2

Rewrite $- 9$ as $16$ plus $- 25$

$20 x^{2} + (16 - 25) x - 20 = 0$

Step 4.3.1.3

Apply the distributive property.

$20 x^{2} + 16 x - 25 x - 20 = 0$

Step 4.3.2

Factor out the greatest common factor from each group.

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Step 4.3.2.1

Group the first two terms and the last two terms.

$(20 x^{2} + 16 x) - 25 x - 20 = 0$

Step 4.3.2.2

Factor out the greatest common factor (GCF) from each group.

$4 x (5 x + 4) - 5 (5 x + 4) = 0$

Step 4.3.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 4$ .

$(5 x + 4) (4 x - 5) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5 x + 4 = 0$

$4 x - 5 = 0$

Step 4.5

Set $5 x + 4$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $5 x + 4$ equal to $0$ .

$5 x + 4 = 0$

Step 4.5.2

Solve $5 x + 4 = 0$ for $x$ .

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Step 4.5.2.1

Subtract $4$ from both sides of the equation.

$5 x = - 4$

Step 4.5.2.2

Divide each term in $5 x = - 4$ by $5$ and simplify.

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Step 4.5.2.2.1

Divide each term in $5 x = - 4$ by $5$ .

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{5}$

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{5}$

Step 4.6

Set $4 x - 5$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Step 4.6.2

Solve $4 x - 5 = 0$ for $x$ .

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Step 4.6.2.1

Add $5$ to both sides of the equation.

$4 x = 5$

Step 4.6.2.2

Divide each term in $4 x = 5$ by $4$ and simplify.

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Step 4.6.2.2.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 4.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 4.7

The final solution is all the values that make $(5 x + 4) (4 x - 5) = 0$ true.

$x = - \frac{4}{5}, \frac{5}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{5}, \frac{5}{4}$

Decimal Form:

$x = - 0.8, 1.25$

Mixed Number Form:

$x = - \frac{4}{5}, 1 \frac{1}{4}$