Finite Math Examples

$8.44 = \frac{3900 x + 1000}{3 x^{2} + 100}$

Step 1

Rewrite the equation as $\frac{3900 x + 1000}{3 x^{2} + 100} = 8.44$ .

$\frac{3900 x + 1000}{3 x^{2} + 100} = 8.44$

Step 2

Factor $100$ out of $3900 x + 1000$ .

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Step 2.1

Factor $100$ out of $3900 x$ .

$\frac{100 (39 x) + 1000}{3 x^{2} + 100} = 8.44$

Step 2.2

Factor $100$ out of $1000$ .

$\frac{100 (39 x) + 100 (10)}{3 x^{2} + 100} = 8.44$

Step 2.3

Factor $100$ out of $100 (39 x) + 100 (10)$ .

$\frac{100 (39 x + 10)}{3 x^{2} + 100} = 8.44$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$3 x^{2} + 100, 1$

Step 3.2

Remove parentheses.

$3 x^{2} + 100, 1$

Step 3.3

The LCM of one and any expression is the expression.

$3 x^{2} + 100$

Step 4

Multiply each term in $\frac{100 (39 x + 10)}{3 x^{2} + 100} = 8.44$ by $3 x^{2} + 100$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{100 (39 x + 10)}{3 x^{2} + 100} = 8.44$ by $3 x^{2} + 100$ .

$\frac{100 (39 x + 10)}{3 x^{2} + 100} (3 x^{2} + 100) = 8.44 (3 x^{2} + 100)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3 x^{2} + 100$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{100 (39 x + 10)}{3 x^{2} + 100} (3 x^{2} + 100) = 8.44 (3 x^{2} + 100)$

Step 4.2.1.2

Rewrite the expression.

$100 (39 x + 10) = 8.44 (3 x^{2} + 100)$

Step 4.2.2

Apply the distributive property.

$100 (39 x) + 100 \cdot 10 = 8.44 (3 x^{2} + 100)$

Step 4.2.3

Multiply.

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Step 4.2.3.1

Multiply $39$ by $100$ .

$3900 x + 100 \cdot 10 = 8.44 (3 x^{2} + 100)$

Step 4.2.3.2

Multiply $100$ by $10$ .

$3900 x + 1000 = 8.44 (3 x^{2} + 100)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$3900 x + 1000 = 8.44 (3 x^{2}) + 8.44 \cdot 100$

Step 4.3.2

Multiply.

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Step 4.3.2.1

Multiply $3$ by $8.44$ .

$3900 x + 1000 = 25.32 x^{2} + 8.44 \cdot 100$

Step 4.3.2.2

Multiply $8.44$ by $100$ .

$3900 x + 1000 = 25.32 x^{2} + 844$

Step 5

Solve the equation.

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Step 5.1

Subtract $25.32 x^{2}$ from both sides of the equation.

$3900 x + 1000 - 25.32 x^{2} = 844$

Step 5.2

Subtract $844$ from both sides of the equation.

$3900 x + 1000 - 25.32 x^{2} - 844 = 0$

Step 5.3

Subtract $844$ from $1000$ .

$3900 x - 25.32 x^{2} + 156 = 0$

Step 5.4

Factor $- 0.12$ out of $3900 x - 25.32 x^{2} + 156$ .

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Step 5.4.1

Reorder $3900 x$ and $- 25.32 x^{2}$ .

$- 25.32 x^{2} + 3900 x + 156 = 0$

Step 5.4.2

Factor $- 0.12$ out of $- 25.32 x^{2}$ .

$- 0.12 (211 x^{2}) + 3900 x + 156 = 0$

Step 5.4.3

Factor $- 0.12$ out of $3900 x$ .

$- 0.12 (211 x^{2}) - 0.12 (- 32500 x) + 156 = 0$

Step 5.4.4

Factor $- 0.12$ out of $156$ .

$- 0.12 (211 x^{2}) - 0.12 (- 32500 x) - 0.12 \cdot - 1300 = 0$

Step 5.4.5

Factor $- 0.12$ out of $- 0.12 (211 x^{2}) - 0.12 (- 32500 x)$ .

$- 0.12 (211 x^{2} - 32500 x) - 0.12 \cdot - 1300 = 0$

Step 5.4.6

Factor $- 0.12$ out of $- 0.12 (211 x^{2} - 32500 x) - 0.12 (- 1300)$ .

$- 0.12 (211 x^{2} - 32500 x - 1300) = 0$

Step 5.5

Divide each term in $- 0.12 (211 x^{2} - 32500 x - 1300) = 0$ by $- 0.12$ and simplify.

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Step 5.5.1

Divide each term in $- 0.12 (211 x^{2} - 32500 x - 1300) = 0$ by $- 0.12$ .

$\frac{- 0.12 (211 x^{2} - 32500 x - 1300)}{- 0.12} = \frac{0}{- 0.12}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- 0.12$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{- 0.12 (211 x^{2} - 32500 x - 1300)}{- 0.12} = \frac{0}{- 0.12}$

Step 5.5.2.1.2

Divide $211 x^{2} - 32500 x - 1300$ by $1$ .

$211 x^{2} - 32500 x - 1300 = \frac{0}{- 0.12}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $0$ by $- 0.12$ .

$211 x^{2} - 32500 x - 1300 = 0$

Step 5.6

Use the quadratic formula to find the solutions.

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

Step 5.7

Substitute the values $a = 211$ , $b = - 32500$ , and $c = - 1300$ into the quadratic formula and solve for $x$ .

$\frac{32500 \pm \sqrt{{(- 32500)}^{2} - 4 \cdot (211 \cdot - 1300)}}{2 \cdot 211}$

Step 5.8

Simplify.

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Step 5.8.1

Simplify the numerator.

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Step 5.8.1.1

Raise $- 32500$ to the power of $2$ .

$x = \frac{32500 \pm \sqrt{1056250000 - 4 \cdot 211 \cdot - 1300}}{2 \cdot 211}$

Step 5.8.1.2

Multiply $- 4 \cdot 211 \cdot - 1300$ .

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Step 5.8.1.2.1

Multiply $- 4$ by $211$ .

$x = \frac{32500 \pm \sqrt{1056250000 - 844 \cdot - 1300}}{2 \cdot 211}$

Step 5.8.1.2.2

Multiply $- 844$ by $- 1300$ .

$x = \frac{32500 \pm \sqrt{1056250000 + 1097200}}{2 \cdot 211}$

Step 5.8.1.3

Add $1056250000$ and $1097200$ .

$x = \frac{32500 \pm \sqrt{1057347200}}{2 \cdot 211}$

Step 5.8.1.4

Rewrite $1057347200$ as $40^{2} \cdot 660842$ .

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Step 5.8.1.4.1

Factor $1600$ out of $1057347200$ .

$x = \frac{32500 \pm \sqrt{1600 (660842)}}{2 \cdot 211}$

Step 5.8.1.4.2

Rewrite $1600$ as $40^{2}$ .

$x = \frac{32500 \pm \sqrt{40^{2} \cdot 660842}}{2 \cdot 211}$

Step 5.8.1.5

Pull terms out from under the radical.

$x = \frac{32500 \pm 40 \sqrt{660842}}{2 \cdot 211}$

Step 5.8.2

Multiply $2$ by $211$ .

$x = \frac{32500 \pm 40 \sqrt{660842}}{422}$

Step 5.8.3

Simplify $\frac{32500 \pm 40 \sqrt{660842}}{422}$ .

$x = \frac{16250 \pm 20 \sqrt{660842}}{211}$

Step 5.9

The final answer is the combination of both solutions.

$x = \frac{16250 + 20 \sqrt{660842}}{211}, \frac{16250 - 20 \sqrt{660842}}{211}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{16250 + 20 \sqrt{660842}}{211}, \frac{16250 - 20 \sqrt{660842}}{211}$

Decimal Form:

$x = 154.06842563 \dots, - 0.03998961 \dots$