Finite Math Examples

$40 x - 39 = \frac{40}{x}$

Step 1

Add $39$ to both sides of the equation.

$40 x = \frac{40}{x} + 39$

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 $40 x = \frac{40}{x} + 39$ by $x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $40 x = \frac{40}{x} + 39$ by $x$ .

$40 x \cdot x = \frac{40}{x} x + 39 x$

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 $x$ .

$40 (x \cdot x) = \frac{40}{x} x + 39 x$

Step 3.2.1.2

Multiply $x$ by $x$ .

$40 x^{2} = \frac{40}{x} x + 39 x$

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.

$40 x^{2} = \frac{40}{x} x + 39 x$

Step 3.3.1.2

Rewrite the expression.

$40 x^{2} = 40 + 39 x$

Step 4

Solve the equation.

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Step 4.1

Subtract $39 x$ from both sides of the equation.

$40 x^{2} - 39 x = 40$

Step 4.2

Subtract $40$ from both sides of the equation.

$40 x^{2} - 39 x - 40 = 0$

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 $a \cdot c = 40 \cdot - 40 = - 1600$ and whose sum is $b = - 39$ .

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Step 4.3.1.1

Factor $- 39$ out of $- 39 x$ .

$40 x^{2} - 39 x - 40 = 0$

Step 4.3.1.2

Rewrite $- 39$ as $25$ plus $- 64$

$40 x^{2} + (25 - 64) x - 40 = 0$

Step 4.3.1.3

Apply the distributive property.

$40 x^{2} + 25 x - 64 x - 40 = 0$

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.

$(40 x^{2} + 25 x) - 64 x - 40 = 0$

Step 4.3.2.2

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

$5 x (8 x + 5) - 8 (8 x + 5) = 0$

Step 4.3.3

Factor the polynomial by factoring out the greatest common factor, $8 x + 5$ .

$(8 x + 5) (5 x - 8) = 0$

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 $0$ .

$8 x + 5 = 0$

$5 x - 8 = 0$

Step 4.5

Set $8 x + 5$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $8 x + 5$ equal to $0$ .

$8 x + 5 = 0$

Step 4.5.2

Solve $8 x + 5 = 0$ for $x$ .

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Step 4.5.2.1

Subtract $5$ from both sides of the equation.

$8 x = - 5$

Step 4.5.2.2

Divide each term in $8 x = - 5$ by $8$ and simplify.

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Step 4.5.2.2.1

Divide each term in $8 x = - 5$ by $8$ .

$\frac{8 x}{8} = \frac{- 5}{8}$

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 $8$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 5}{8}$

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{8}$

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.

$x = - \frac{5}{8}$

Step 4.6

Set $5 x - 8$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $5 x - 8$ equal to $0$ .

$5 x - 8 = 0$

Step 4.6.2

Solve $5 x - 8 = 0$ for $x$ .

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Step 4.6.2.1

Add $8$ to both sides of the equation.

$5 x = 8$

Step 4.6.2.2

Divide each term in $5 x = 8$ by $5$ and simplify.

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Step 4.6.2.2.1

Divide each term in $5 x = 8$ by $5$ .

$\frac{5 x}{5} = \frac{8}{5}$

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 $5$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{8}{5}$

Step 4.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{5}$

Step 4.7

The final solution is all the values that make $(8 x + 5) (5 x - 8) = 0$ true.

$x = - \frac{5}{8}, \frac{8}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{8}, \frac{8}{5}$

Decimal Form:

$x = - 0.625, 1.6$

Mixed Number Form:

$x = - \frac{5}{8}, 1 \frac{3}{5}$