Finite Math Examples

$- 0.000047 S^{2} + 0.081065 S - 27.895165 = 0$

Step 1

Add $27.895165$ to both sides of the equation.

$- 0.000047 S^{2} + 0.081065 S = 27.895165$

Step 2

Divide each term in $- 0.000047 S^{2} + 0.081065 S = 27.895165$ by $- 0.000047$ and simplify.

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Step 2.1

Divide each term in $- 0.000047 S^{2} + 0.081065 S = 27.895165$ by $- 0.000047$ .

$\frac{- 0.000047 S^{2}}{- 0.000047} + \frac{0.081065 S}{- 0.000047} = \frac{27.895165}{- 0.000047}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $- 0.000047$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{- 0.000047 S^{2}}{- 0.000047} + \frac{0.081065 S}{- 0.000047} = \frac{27.895165}{- 0.000047}$

Step 2.2.1.1.2

Divide $S^{2}$ by $1$ .

$S^{2} + \frac{0.081065 S}{- 0.000047} = \frac{27.895165}{- 0.000047}$

Step 2.2.1.2

Move the negative in front of the fraction.

$S^{2} - \frac{0.081065 S}{0.000047} = \frac{27.895165}{- 0.000047}$

Step 2.2.1.3

Factor $0.081065$ out of $0.081065 S$ .

$S^{2} - \frac{0.081065 (S)}{0.000047} = \frac{27.895165}{- 0.000047}$

Step 2.2.1.4

Factor $0.000047$ out of $0.000047$ .

$S^{2} - \frac{0.081065 (S)}{0.000047 (1)} = \frac{27.895165}{- 0.000047}$

Step 2.2.1.5

Separate fractions.

$S^{2} - (\frac{0.081065}{0.000047} \cdot \frac{S}{1}) = \frac{27.895165}{- 0.000047}$

Step 2.2.1.6

Divide $0.081065$ by $0.000047$ .

$S^{2} - (1724.78723404 \frac{S}{1}) = \frac{27.895165}{- 0.000047}$

Step 2.2.1.7

Divide $S$ by $1$ .

$S^{2} - (1724.78723404 S) = \frac{27.895165}{- 0.000047}$

Step 2.2.1.8

Multiply $1724.78723404$ by $- 1$ .

$S^{2} - 1724.78723404 S = \frac{27.895165}{- 0.000047}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $27.895165$ by $- 0.000047$ .

$S^{2} - 1724.78723404 S = - 593514.14893617$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{81065}{94})}^{2}$

Step 4

Add the term to each side of the equation.

$S^{2} - 1724.78723404 S + {(- \frac{81065}{94})}^{2} = - 593514.14893617 + {(- \frac{81065}{94})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1.1

Apply the product rule to $- \frac{81065}{94}$ .

$S^{2} - 1724.78723404 S + {(- 1)}^{2} {(\frac{81065}{94})}^{2} = - 593514.14893617 + {(- \frac{81065}{94})}^{2}$

Step 5.1.1.1.2

Apply the product rule to $\frac{81065}{94}$ .

$S^{2} - 1724.78723404 S + {(- 1)}^{2} \frac{81065^{2}}{94^{2}} = - 593514.14893617 + {(- \frac{81065}{94})}^{2}$

Step 5.1.1.2

Raise $- 1$ to the power of $2$ .

$S^{2} - 1724.78723404 S + 1 \frac{81065^{2}}{94^{2}} = - 593514.14893617 + {(- \frac{81065}{94})}^{2}$

Step 5.1.1.3

Multiply $\frac{81065^{2}}{94^{2}}$ by $1$ .

$S^{2} - 1724.78723404 S + \frac{81065^{2}}{94^{2}} = - 593514.14893617 + {(- \frac{81065}{94})}^{2}$

Step 5.1.1.4

Raise $81065$ to the power of $2$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{94^{2}} = - 593514.14893617 + {(- \frac{81065}{94})}^{2}$

Step 5.1.1.5

Raise $94$ to the power of $2$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 + {(- \frac{81065}{94})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 593514.14893617 + {(- \frac{81065}{94})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{81065}{94}$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 + {(- 1)}^{2} {(\frac{81065}{94})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to $\frac{81065}{94}$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 + {(- 1)}^{2} \frac{81065^{2}}{94^{2}}$

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 + 1 \frac{81065^{2}}{94^{2}}$

Step 5.2.1.1.3

Multiply $\frac{81065^{2}}{94^{2}}$ by $1$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 + \frac{81065^{2}}{94^{2}}$

Step 5.2.1.1.4

Raise $81065$ to the power of $2$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 + \frac{6571534225}{94^{2}}$

Step 5.2.1.1.5

Raise $94$ to the power of $2$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 + \frac{6571534225}{8836}$

Step 5.2.1.2

To write $- 593514.14893617$ as a fraction with a common denominator, multiply by $\frac{8836}{8836}$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = - 593514.14893617 \cdot \frac{8836}{8836} + \frac{6571534225}{8836}$

Step 5.2.1.3

Combine $- 593514.14893617$ and $\frac{8836}{8836}$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{- 593514.14893617 \cdot 8836}{8836} + \frac{6571534225}{8836}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{- 593514.14893617 \cdot 8836 + 6571534225}{8836}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $- 593514.14893617$ by $8836$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{- 5244291020 + 6571534225}{8836}$

Step 5.2.1.5.2

Add $- 5244291020$ and $6571534225$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{1327243205}{8836}$

Step 5.2.1.6

Cancel the common factor of $1327243205$ and $8836$ .

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Step 5.2.1.6.1

Rewrite $1327243205$ as $1 (1327243205)$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{1 (1327243205)}{8836}$

Step 5.2.1.6.2

Cancel the common factors.

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Step 5.2.1.6.2.1

Rewrite $8836$ as $1 (8836)$ .

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{1 \cdot 1327243205}{1 \cdot 8836}$

Step 5.2.1.6.2.2

Cancel the common factor.

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{1 \cdot 1327243205}{1 \cdot 8836}$

Step 5.2.1.6.2.3

Rewrite the expression.

$S^{2} - 1724.78723404 S + \frac{6571534225}{8836} = \frac{1327243205}{8836}$

Step 6

Factor the perfect trinomial square into ${(S - \frac{81065}{94})}^{2}$ .

${(S - \frac{81065}{94})}^{2} = \frac{1327243205}{8836}$

Step 7

Solve the equation for $S$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$S - \frac{81065}{94} = \pm \sqrt{\frac{1327243205}{8836}}$

Step 7.2

Simplify $\pm \sqrt{\frac{1327243205}{8836}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{1327243205}{8836}}$ as $\frac{\sqrt{1327243205}}{\sqrt{8836}}$ .

$S - \frac{81065}{94} = \pm \frac{\sqrt{1327243205}}{\sqrt{8836}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Rewrite $8836$ as $94^{2}$ .

$S - \frac{81065}{94} = \pm \frac{\sqrt{1327243205}}{\sqrt{94^{2}}}$

Step 7.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$S - \frac{81065}{94} = \pm \frac{\sqrt{1327243205}}{94}$

Step 7.3

Add $\frac{81065}{94}$ to both sides of the equation.

$S = \pm \frac{\sqrt{1327243205}}{94} + \frac{81065}{94}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$S = \pm \frac{\sqrt{1327243205}}{94} + \frac{81065}{94}$

Decimal Form:

$S = 1249.96116177 \dots, 474.82607227 \dots$