Finite Math Examples

$\frac{x^{2}}{3.4 \cdot 10^{- 3} - x} = 1.4 \cdot 10^{- 4}$

Step 1

Factor each term.

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Step 1.1

Factor $0.2$ out of $3.4 \cdot 10^{- 3} - x$ .

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Step 1.1.1

Factor $0.2$ out of $3.4 \cdot 10^{- 3}$ .

$\frac{x^{2}}{0.2 \cdot 17 \cdot 10^{- 3} - x} = 1.4 \cdot 10^{- 4}$

Step 1.1.2

Factor $0.2$ out of $- x$ .

$\frac{x^{2}}{0.2 \cdot 17 \cdot 10^{- 3} + 0.2 (- 5 x)} = 1.4 \cdot 10^{- 4}$

Step 1.1.3

Factor $0.2$ out of $0.2 \cdot 17 \cdot 10^{- 3} + 0.2 (- 5 x)$ .

$\frac{x^{2}}{0.2 \cdot (17 \cdot 10^{- 3} - 5 x)} = 1.4 \cdot 10^{- 4}$

Step 1.1.4

Multiply $0.2$ by $17 \cdot 10^{- 3} - 5 x$ .

$\frac{x^{2}}{0.2 (17 \cdot 10^{- 3} - 5 x)} = 1.4 \cdot 10^{- 4}$

Step 1.2

Move the decimal point in $17$ to the left by $1$ place and increase the power of $10^{- 3}$ by $1$ .

$\frac{x^{2}}{0.2 (1.7 \cdot 10^{- 2} - 5 x)} = 1.4 \cdot 10^{- 4}$

Step 1.3

Factor.

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Step 1.3.1

Rewrite $1.7 \cdot 10^{- 2} - 5 x$ in a factored form.

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Step 1.3.1.1

Factor $0.1$ out of $1.7 \cdot 10^{- 2} - 5 x$ .

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Step 1.3.1.1.1

Factor $0.1$ out of $1.7 \cdot 10^{- 2}$ .

$\frac{x^{2}}{0.2 (0.1 \cdot 17 \cdot 10^{- 2} - 5 x)} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.1.2

Factor $0.1$ out of $- 5 x$ .

$\frac{x^{2}}{0.2 (0.1 \cdot 17 \cdot 10^{- 2} + 0.1 (- 50 x))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.1.3

Factor $0.1$ out of $0.1 \cdot 17 \cdot 10^{- 2} + 0.1 (- 50 x)$ .

$\frac{x^{2}}{0.2 (0.1 (17 \cdot 10^{- 2} - 50 x))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.2

Move the decimal point in $17$ to the left by $1$ place and increase the power of $10^{- 2}$ by $1$ .

$\frac{x^{2}}{0.2 (0.1 (1.7 \cdot 10^{- 1} - 50 x))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3

Factor.

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Step 1.3.1.3.1

Rewrite $1.7 \cdot 10^{- 1} - 50 x$ in a factored form.

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Step 1.3.1.3.1.1

Factor $0.1$ out of $1.7 \cdot 10^{- 1} - 50 x$ .

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Step 1.3.1.3.1.1.1

Factor $0.1$ out of $1.7 \cdot 10^{- 1}$ .

$\frac{x^{2}}{0.2 (0.1 (0.1 \cdot 17 \cdot 10^{- 1} - 50 x))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.1.2

Factor $0.1$ out of $- 50 x$ .

$\frac{x^{2}}{0.2 (0.1 (0.1 \cdot 17 \cdot 10^{- 1} + 0.1 (- 500 x)))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.1.3

Factor $0.1$ out of $0.1 \cdot 17 \cdot 10^{- 1} + 0.1 (- 500 x)$ .

$\frac{x^{2}}{0.2 (0.1 (0.1 (17 \cdot 10^{- 1} - 500 x)))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.2

Move the decimal point in $17$ to the left by $1$ place and increase the power of $10^{- 1}$ by $1$ .

$\frac{x^{2}}{0.2 (0.1 (0.1 (1.7 \cdot 10^{0} - 500 x)))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.3

Convert $1.7 \cdot 10^{0}$ from scientific notation.

$\frac{x^{2}}{0.2 (0.1 (0.1 (1.7 - 500 x)))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.4

Factor.

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Step 1.3.1.3.1.4.1

Factor $0.1$ out of $1.7 - 500 x$ .

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Step 1.3.1.3.1.4.1.1

Factor $0.1$ out of $1.7$ .

$\frac{x^{2}}{0.2 (0.1 (0.1 (0.1 (17) - 500 x)))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.4.1.2

Factor $0.1$ out of $- 500 x$ .

$\frac{x^{2}}{0.2 (0.1 (0.1 (0.1 (17) + 0.1 (- 5000 x))))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.4.1.3

Factor $0.1$ out of $0.1 (17) + 0.1 (- 5000 x)$ .

$\frac{x^{2}}{0.2 (0.1 (0.1 (0.1 (17 - 5000 x))))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.4.2

Remove unnecessary parentheses.

$\frac{x^{2}}{0.2 (0.1 (0.1 \cdot 0.1 (17 - 5000 x)))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.1.5

Multiply $0.1$ by $0.1$ .

$\frac{x^{2}}{0.2 (0.1 (0.01 (17 - 5000 x)))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.3.2

Remove unnecessary parentheses.

$\frac{x^{2}}{0.2 (0.1 \cdot 0.01 (17 - 5000 x))} = 1.4 \cdot 10^{- 4}$

Step 1.3.1.4

Multiply $0.1$ by $0.01$ .

$\frac{x^{2}}{0.2 (0.001 (17 - 5000 x))} = 1.4 \cdot 10^{- 4}$

Step 1.3.2

Remove unnecessary parentheses.

$\frac{x^{2}}{0.2 \cdot 0.001 (17 - 5000 x)} = 1.4 \cdot 10^{- 4}$

Step 1.4

Multiply $0.2$ by $0.001$ .

$\frac{x^{2}}{0.0002 (17 - 5000 x)} = 1.4 \cdot 10^{- 4}$

Step 1.5

Multiply by $1$ .

$\frac{1 x^{2}}{0.0002 (17 - 5000 x)} = 1.4 \cdot 10^{- 4}$

Step 1.6

Separate fractions.

$\frac{1}{0.0002} \cdot \frac{x^{2}}{17 - 5000 x} = 1.4 \cdot 10^{- 4}$

Step 1.7

Divide $1$ by $0.0002$ .

$5000 \frac{x^{2}}{17 - 5000 x} = 1.4 \cdot 10^{- 4}$

Step 1.8

Combine $5000$ and $\frac{x^{2}}{17 - 5000 x}$ .

$\frac{5000 x^{2}}{17 - 5000 x} = 1.4 \cdot 10^{- 4}$

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.

$17 - 5000 x, 1, 1$

Step 2.2

Remove parentheses.

$17 - 5000 x, 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$17 - 5000 x$

Step 3

Multiply each term in $\frac{5000 x^{2}}{17 - 5000 x} = 1.4 \cdot 10^{- 4}$ by $17 - 5000 x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{5000 x^{2}}{17 - 5000 x} = 1.4 \cdot 10^{- 4}$ by $17 - 5000 x$ .

$\frac{5000 x^{2}}{17 - 5000 x} (17 - 5000 x) = 1.4 \cdot 10^{- 4} (17 - 5000 x)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $17 - 5000 x$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5000 x^{2}}{17 - 5000 x} (17 - 5000 x) = 1.4 \cdot 10^{- 4} (17 - 5000 x)$

Step 3.2.1.2

Rewrite the expression.

$5000 x^{2} = 1.4 \cdot 10^{- 4} (17 - 5000 x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify by multiplying through.

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Step 3.3.1.1

Apply the distributive property.

$5000 x^{2} = 1.4 \cdot 10^{- 4} \cdot 17 + 1.4 \cdot 10^{- 4} (- 5000 x)$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Multiply $1.4$ by $17$ .

$5000 x^{2} = 23.8 \cdot 10^{- 4} + 1.4 \cdot 10^{- 4} (- 5000 x)$

Step 3.3.1.2.2

Multiply $- 5000$ by $1.4$ .

$5000 x^{2} = 23.8 \cdot 10^{- 4} - 7000 \cdot 10^{- 4} x$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Move the decimal point in $23.8$ to the left by $1$ place and increase the power of $10^{- 4}$ by $1$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} - 7000 \cdot 10^{- 4} x$

Step 3.3.2.2

Move the decimal point in $- 7000$ to the left by $3$ places and increase the power of $10^{- 4}$ by $3$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} - 7 \cdot 10^{- 1} x$

Step 3.3.3

Reorder factors in $2.38 \cdot 10^{- 3} - 7 \cdot 10^{- 1} x$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} - 7 x \cdot 10^{- 1}$

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 $b^{- n} = \frac{1}{b^{n}}$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} - 7 x \cdot \frac{1}{10}$

Step 4.1.2

Multiply $- 7 x \frac{1}{10}$ .

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Step 4.1.2.1

Combine $- 7$ and $\frac{1}{10}$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} + x \frac{- 7}{10}$

Step 4.1.2.2

Combine $x$ and $\frac{- 7}{10}$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} + \frac{x \cdot - 7}{10}$

Step 4.1.3

Cancel the common factor of $- 7$ and $10$ .

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Step 4.1.3.1

Factor $1$ out of $x \cdot - 7$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} + \frac{1 (x \cdot - 7)}{10}$

Step 4.1.3.2

Cancel the common factors.

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Step 4.1.3.2.1

Rewrite $10$ as $1 (10)$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} + \frac{1 (x \cdot - 7)}{1 (10)}$

Step 4.1.3.2.2

Cancel the common factor.

$5000 x^{2} = 2.38 \cdot 10^{- 3} + \frac{1 (x \cdot - 7)}{1 \cdot 10}$

Step 4.1.3.2.3

Rewrite the expression.

$5000 x^{2} = 2.38 \cdot 10^{- 3} + \frac{x \cdot - 7}{10}$

Step 4.1.4

Move $- 7$ to the left of $x$ .

$5000 x^{2} = 2.38 \cdot 10^{- 3} + \frac{- 7 \cdot x}{10}$

Step 4.1.5

Move the negative in front of the fraction.

$5000 x^{2} = 2.38 \cdot 10^{- 3} - \frac{7 x}{10}$

Step 4.2

Add $\frac{7 x}{10}$ to both sides of the equation.

$5000 x^{2} + \frac{7 x}{10} = 2.38 \cdot 10^{- 3}$

Step 4.3

Subtract $2.38 \cdot 10^{- 3}$ from both sides of the equation.

$5000 x^{2} + \frac{7 x}{10} - 2.38 \cdot 10^{- 3} = 0$

Step 4.4

Multiply through by the least common denominator $10$ , then simplify.

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Step 4.4.1

Apply the distributive property.

$10 (5000 x^{2}) + 10 (\frac{7 x}{10}) + 10 \cdot - 2.38 \cdot 10^{- 3} = 0$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Multiply $5000$ by $10$ .

$50000 x^{2} + 10 (\frac{7 x}{10}) + 10 \cdot - 2.38 \cdot 10^{- 3} = 0$

Step 4.4.2.2

Cancel the common factor of $10$ .

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Step 4.4.2.2.1

Cancel the common factor.

$50000 x^{2} + 10 (\frac{7 x}{10}) + 10 \cdot - 2.38 \cdot 10^{- 3} = 0$

Step 4.4.2.2.2

Rewrite the expression.

$50000 x^{2} + 7 x + 10 \cdot - 2.38 \cdot 10^{- 3} = 0$

Step 4.4.2.3

Multiply $10$ by $- 2.38$ .

$50000 x^{2} + 7 x - 23.8 \cdot 10^{- 3} = 0$

Step 4.4.3

Move the decimal point in $- 23.8$ to the left by $1$ place and increase the power of $10^{- 3}$ by $1$ .

$50000 x^{2} + 7 x - 2.38 \cdot 10^{- 2} = 0$

Step 4.5

Use the quadratic formula to find the solutions.

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

Step 4.6

Substitute the values $a = 50000$ , $b = 7$ , and $c = - 2.38 \cdot 10^{- 2}$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (50000 \cdot - 2.38 \cdot 10^{- 2})}}{2 \cdot 50000}$

Step 4.7

Simplify.

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Step 4.7.1

Multiply $50000$ by $- 2.38$ .

$x = \frac{- 7 \pm \sqrt{7^{2} - 4 \cdot - 119000 \cdot 10^{- 2}}}{2 \cdot 50000}$

Step 4.7.2

Multiply $- 4$ by $- 119000$ .

$x = \frac{- 7 \pm \sqrt{7^{2} + 476000 \cdot 10^{- 2}}}{2 \cdot 50000}$

Step 4.7.3

Simplify the numerator.

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Step 4.7.3.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 + 476000 \cdot 10^{- 2}}}{2 \cdot 50000}$

Step 4.7.3.2

Move the decimal point in $476000$ to the left by $5$ places and increase the power of $10^{- 2}$ by $5$ .

$x = \frac{- 7 \pm \sqrt{49 + 4.76 \cdot 10^{3}}}{2 \cdot 50000}$

Step 4.7.3.3

Convert $49$ to scientific notation.

$x = \frac{- 7 \pm \sqrt{4.9 \cdot 10 + 4.76 \cdot 10^{3}}}{2 \cdot 50000}$

Step 4.7.3.4

Move the decimal point in $4.9$ to the left by $2$ places and increase the power of $10^{1}$ by $2$ .

$x = \frac{- 7 \pm \sqrt{0.049 \cdot 10^{3} + 4.76 \cdot 10^{3}}}{2 \cdot 50000}$

Step 4.7.3.5

Factor $10^{3}$ out of $0.049 \cdot 10^{3} + 4.76 \cdot 10^{3}$ .

$x = \frac{- 7 \pm \sqrt{(0.049 + 4.76) \cdot 10^{3}}}{2 \cdot 50000}$

Step 4.7.3.6

Add $0.049$ and $4.76$ .

$x = \frac{- 7 \pm \sqrt{4.809 \cdot 10^{3}}}{2 \cdot 50000}$

Step 4.7.3.7

Raise $10$ to the power of $3$ .

$x = \frac{- 7 \pm \sqrt{4.809 \cdot 1000}}{2 \cdot 50000}$

Step 4.7.3.8

Multiply $4.809$ by $1000$ .

$x = \frac{- 7 \pm \sqrt{4809}}{2 \cdot 50000}$

Step 4.7.4

Multiply $2$ by $50000$ .

$x = \frac{- 7 \pm \sqrt{4809}}{100000}$

Step 4.8

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{4809}}{100000}, - \frac{7 + \sqrt{4809}}{100000}$

$x = \frac{- 7 \pm \sqrt{4809}}{100000}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 7 \pm \sqrt{4809}}{100000}$

Decimal Form:

$x = 0.00062346 \dots, - 0.00076346 \dots$

Step 6