Finite Math Examples

$\frac{x^{2}}{0.05 - x} = 3.1 \cdot 10^{- 8}$

Step 1

Factor each term.

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Step 1.1

Factor $0.05$ out of $0.05 - x$ .

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Step 1.1.1

Factor $0.05$ out of $0.05$ .

$\frac{x^{2}}{0.05 (1) - x} = 3.1 \cdot 10^{- 8}$

Step 1.1.2

Factor $0.05$ out of $- x$ .

$\frac{x^{2}}{0.05 (1) + 0.05 (- 20 x)} = 3.1 \cdot 10^{- 8}$

Step 1.1.3

Factor $0.05$ out of $0.05 (1) + 0.05 (- 20 x)$ .

$\frac{x^{2}}{0.05 (1 - 20 x)} = 3.1 \cdot 10^{- 8}$

Step 1.2

Multiply by $1$ .

$\frac{1 x^{2}}{0.05 (1 - 20 x)} = 3.1 \cdot 10^{- 8}$

Step 1.3

Separate fractions.

$\frac{1}{0.05} \cdot \frac{x^{2}}{1 - 20 x} = 3.1 \cdot 10^{- 8}$

Step 1.4

Divide $1$ by $0.05$ .

$20 \frac{x^{2}}{1 - 20 x} = 3.1 \cdot 10^{- 8}$

Step 1.5

Combine $20$ and $\frac{x^{2}}{1 - 20 x}$ .

$\frac{20 x^{2}}{1 - 20 x} = 3.1 \cdot 10^{- 8}$

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.

$1 - 20 x, 1, 1$

Step 2.2

Remove parentheses.

$1 - 20 x, 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$1 - 20 x$

Step 3

Multiply each term in $\frac{20 x^{2}}{1 - 20 x} = 3.1 \cdot 10^{- 8}$ by $1 - 20 x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{20 x^{2}}{1 - 20 x} = 3.1 \cdot 10^{- 8}$ by $1 - 20 x$ .

$\frac{20 x^{2}}{1 - 20 x} (1 - 20 x) = 3.1 \cdot 10^{- 8} (1 - 20 x)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $1 - 20 x$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{20 x^{2}}{1 - 20 x} (1 - 20 x) = 3.1 \cdot 10^{- 8} (1 - 20 x)$

Step 3.2.1.2

Rewrite the expression.

$20 x^{2} = 3.1 \cdot 10^{- 8} (1 - 20 x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$20 x^{2} = 3.1 \cdot 10^{- 8} \cdot 1 + 3.1 \cdot 10^{- 8} (- 20 x)$

Step 3.3.2

Multiply.

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Step 3.3.2.1

Multiply $3.1$ by $1$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} + 3.1 \cdot 10^{- 8} (- 20 x)$

Step 3.3.2.2

Multiply $- 20$ by $3.1$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - 62 \cdot 10^{- 8} x$

Step 3.3.3

Move the decimal point in $- 62$ to the left by $1$ place and increase the power of $10^{- 8}$ by $1$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - 6.2 \cdot 10^{- 7} x$

Step 3.3.4

Reorder factors in $3.1 \cdot 10^{- 8} - 6.2 \cdot 10^{- 7} x$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - 6.2 x \cdot 10^{- 7}$

Step 4

Solve the equation.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - 6.2 x \cdot \frac{1}{10^{7}}$

Step 4.1.2

Raise $10$ to the power of $7$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - 6.2 x \cdot \frac{1}{10000000}$

Step 4.1.3

Multiply $- 6.2 x \frac{1}{10000000}$ .

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Step 4.1.3.1

Combine $- 6.2$ and $\frac{1}{10000000}$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} + x \frac{- 6.2}{10000000}$

Step 4.1.3.2

Combine $x$ and $\frac{- 6.2}{10000000}$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} + \frac{x \cdot - 6.2}{10000000}$

Step 4.1.4

Move $- 6.2$ to the left of $x$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} + \frac{- 6.2 \cdot x}{10000000}$

Step 4.1.5

Move the negative in front of the fraction.

$20 x^{2} = 3.1 \cdot 10^{- 8} - \frac{6.2 \cdot x}{10000000}$

Step 4.1.6

Factor $6.2$ out of $6.2 \cdot x$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - \frac{6.2 (x)}{10000000}$

Step 4.1.7

Factor $10000000$ out of $10000000$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - \frac{6.2 (x)}{10000000 (1)}$

Step 4.1.8

Separate fractions.

$20 x^{2} = 3.1 \cdot 10^{- 8} - (\frac{6.2}{10000000} \cdot \frac{x}{1})$

Step 4.1.9

Divide $6.2$ by $10000000$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - (0.00000062 \frac{x}{1})$

Step 4.1.10

Divide $x$ by $1$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - (0.00000062 x)$

Step 4.1.11

Multiply $0.00000062$ by $- 1$ .

$20 x^{2} = 3.1 \cdot 10^{- 8} - 0.00000062 x$

Step 4.2

Add $0.00000062 x$ to both sides of the equation.

$20 x^{2} + 0.00000062 x = 3.1 \cdot 10^{- 8}$

Step 4.3

Subtract $3.1 \cdot 10^{- 8}$ from both sides of the equation.

$20 x^{2} + 0.00000062 x - 3.1 \cdot 10^{- 8} = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $0.00000002$ out of $20 x^{2} + 0.00000062 x - 3.1 \cdot 10^{- 8}$ .

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Step 4.4.1.1

Factor $0.00000002$ out of $20 x^{2}$ .

$0.00000002 (1000000000 x^{2}) + 0.00000062 x - 3.1 \cdot 10^{- 8} = 0$

Step 4.4.1.2

Factor $0.00000002$ out of $0.00000062 x$ .

$0.00000002 (1000000000 x^{2}) + 0.00000002 (31 x) - 3.1 \cdot 10^{- 8} = 0$

Step 4.4.1.3

Factor $0.00000002$ out of $- 3.1 \cdot 10^{- 8}$ .

$0.00000002 (1000000000 x^{2}) + 0.00000002 (31 x) + 0.00000002 \cdot - 155000000 \cdot 10^{- 8} = 0$

Step 4.4.1.4

Factor $0.00000002$ out of $0.00000002 (1000000000 x^{2}) + 0.00000002 (31 x)$ .

$0.00000002 (1000000000 x^{2} + 31 x) + 0.00000002 \cdot - 155000000 \cdot 10^{- 8} = 0$

Step 4.4.1.5

Factor $0.00000002$ out of $0.00000002 (1000000000 x^{2} + 31 x) + 0.00000002 \cdot - 155000000 \cdot 10^{- 8}$ .

$0.00000002 (1000000000 x^{2} + 31 x - 155000000 \cdot 10^{- 8}) = 0$

Step 4.4.2

Move the decimal point in $- 155000000$ to the left by $8$ places and increase the power of $10^{- 8}$ by $8$ .

$0.00000002 (1000000000 x^{2} + 31 x - 1.55 \cdot 10^{0}) = 0$

Step 4.4.3

Convert $- 1.55 \cdot 10^{0}$ from scientific notation.

$0.00000002 (1000000000 x^{2} + 31 x - 1.55) = 0$

Step 4.5

Divide each term in $0.00000002 (1000000000 x^{2} + 31 x - 1.55) = 0$ by $0.00000002$ and simplify.

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Step 4.5.1

Divide each term in $0.00000002 (1000000000 x^{2} + 31 x - 1.55) = 0$ by $0.00000002$ .

$\frac{0.00000002 (1000000000 x^{2} + 31 x - 1.55)}{0.00000002} = \frac{0}{0.00000002}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $0.00000002$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{0.00000002 (1000000000 x^{2} + 31 x - 1.55)}{0.00000002} = \frac{0}{0.00000002}$

Step 4.5.2.1.2

Divide $1000000000 x^{2} + 31 x - 1.55$ by $1$ .

$1000000000 x^{2} + 31 x - 1.55 = \frac{0}{0.00000002}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Divide $0$ by $0.00000002$ .

$1000000000 x^{2} + 31 x - 1.55 = 0$

Step 4.6

Use the quadratic formula to find the solutions.

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

Step 4.7

Substitute the values $a = 1000000000$ , $b = 31$ , and $c = - 1.55$ into the quadratic formula and solve for $x$ .

$\frac{- 31 \pm \sqrt{31^{2} - 4 \cdot (1000000000 \cdot - 1.55)}}{2 \cdot 1000000000}$

Step 4.8

Simplify.

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Step 4.8.1

Simplify the numerator.

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Step 4.8.1.1

Raise $31$ to the power of $2$ .

$x = \frac{- 31 \pm \sqrt{961 - 4 \cdot 1000000000 \cdot - 1.55}}{2 \cdot 1000000000}$

Step 4.8.1.2

Multiply $- 4 \cdot 1000000000 \cdot - 1.55$ .

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Step 4.8.1.2.1

Multiply $- 4$ by $1000000000$ .

$x = \frac{- 31 \pm \sqrt{961 - 4000000000 \cdot - 1.55}}{2 \cdot 1000000000}$

Step 4.8.1.2.2

Multiply $- 4000000000$ by $- 1.55$ .

$x = \frac{- 31 \pm \sqrt{961 + 6200000000}}{2 \cdot 1000000000}$

Step 4.8.1.3

Add $961$ and $6200000000$ .

$x = \frac{- 31 \pm \sqrt{6200000961}}{2 \cdot 1000000000}$

Step 4.8.2

Multiply $2$ by $1000000000$ .

$x = \frac{- 31 \pm \sqrt{6200000961}}{2000000000}$

Step 4.9

The final answer is the combination of both solutions.

$x = - \frac{31 - \sqrt{6200000961}}{2000000000}, - \frac{31 + \sqrt{6200000961}}{2000000000}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{31 - \sqrt{6200000961}}{2000000000}, - \frac{31 + \sqrt{6200000961}}{2000000000}$

Decimal Form:

$x = 3.93545424 \cdot 10^{- 5}, - 3.93855424 \cdot 10^{- 5}$