Finite Math Examples

$0 = 3.51 x^{2} + 24.4 x - (8 \cdot 10^{12})$

Step 1

Rewrite the equation as $3.51 x^{2} + 24.4 x - 1 \cdot 8 \cdot 10^{12} = 0$ .

$3.51 x^{2} + 24.4 x - 1 \cdot 8 \cdot 10^{12} = 0$

Step 2

Multiply $- 1$ by $8$ .

$3.51 x^{2} + 24.4 x - 8 \cdot 10^{12} = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 0.01$ out of $3.51 x^{2} + 24.4 x - 8 \cdot 10^{12}$ .

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Step 3.1.1

Factor $- 0.01$ out of $3.51 x^{2}$ .

$- 0.01 (- 351 x^{2}) + 24.4 x - 8 \cdot 10^{12} = 0$

Step 3.1.2

Factor $- 0.01$ out of $24.4 x$ .

$- 0.01 (- 351 x^{2}) - 0.01 (- 2440 x) - 8 \cdot 10^{12} = 0$

Step 3.1.3

Factor $- 0.01$ out of $- 8 \cdot 10^{12}$ .

$- 0.01 (- 351 x^{2}) - 0.01 (- 2440 x) - 0.01 \cdot 800 \cdot 10^{12} = 0$

Step 3.1.4

Factor $- 0.01$ out of $- 0.01 (- 351 x^{2}) - 0.01 (- 2440 x)$ .

$- 0.01 (- 351 x^{2} - 2440 x) - 0.01 \cdot 800 \cdot 10^{12} = 0$

Step 3.1.5

Factor $- 0.01$ out of $- 0.01 (- 351 x^{2} - 2440 x) - 0.01 \cdot 800 \cdot 10^{12}$ .

$- 0.01 (- 351 x^{2} - 2440 x + 800 \cdot 10^{12}) = 0$

Step 3.2

Move the decimal point in $800$ to the left by $2$ places and increase the power of $10^{12}$ by $2$ .

$- 0.01 (- 351 x^{2} - 2440 x + 8 \cdot 10^{14}) = 0$

Step 3.3

Reorder terms.

$- 0.01 (8 \cdot 10^{14} - 351 x^{2} - 2440 x) = 0$

Step 3.4

Factor.

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Step 3.4.1

Factor $1$ out of $8 \cdot 10^{14} - 351 x^{2} - 2440 x$ .

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Step 3.4.1.1

Factor $1$ out of $8 \cdot 10^{14}$ .

$- 0.01 (1 \cdot 8 \cdot 10^{14} - 351 x^{2} - 2440 x) = 0$

Step 3.4.1.2

Factor $1$ out of $- 351 x^{2}$ .

$- 0.01 (1 \cdot 8 \cdot 10^{14} + 1 (- 351 x^{2}) - 2440 x) = 0$

Step 3.4.1.3

Factor $1$ out of $- 2440 x$ .

$- 0.01 (1 \cdot 8 \cdot 10^{14} + 1 (- 351 x^{2}) + 1 (- 2440 x)) = 0$

Step 3.4.1.4

Factor $1$ out of $1 \cdot 8 \cdot 10^{14} + 1 (- 351 x^{2})$ .

$- 0.01 (1 \cdot (8 \cdot 10^{14} - 351 x^{2}) + 1 (- 2440 x)) = 0$

Step 3.4.1.5

Factor $1$ out of $1 \cdot (8 \cdot 10^{14} - 351 x^{2}) + 1 (- 2440 x)$ .

$- 0.01 (1 (8 \cdot 10^{14} - 351 x^{2} - 2440 x)) = 0$

Step 3.4.2

Remove unnecessary parentheses.

$- 0.01 \cdot (1 (8 \cdot 10^{14} - 351 x^{2} - 2440 x)) = 0$

Step 3.5

Multiply $- 0.01$ by $1$ .

$- 0.01 (8 \cdot 10^{14} - 351 x^{2} - 2440 x) = 0$

Step 4

Divide each term in $- 0.01 (8 \cdot 10^{14} - 351 x^{2} - 2440 x) = 0$ by $- 0.01$ and simplify.

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Step 4.1

Divide each term in $- 0.01 (8 \cdot 10^{14} - 351 x^{2} - 2440 x) = 0$ by $- 0.01$ .

$\frac{- 0.01 (8 \cdot 10^{14} - 351 x^{2} - 2440 x)}{- 0.01} = \frac{0}{- 0.01}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 0.01$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 0.01 (8 \cdot 10^{14} - 351 x^{2} - 2440 x)}{- 0.01} = \frac{0}{- 0.01}$

Step 4.2.1.2

Divide $8 \cdot 10^{14} - 351 x^{2} - 2440 x$ by $1$ .

$8 \cdot 10^{14} - 351 x^{2} - 2440 x = \frac{0}{- 0.01}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 0.01$ .

$8 \cdot 10^{14} - 351 x^{2} - 2440 x = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 351$ , $b = - 2440$ , and $c = 8 \cdot 10^{14}$ into the quadratic formula and solve for $x$ .

$\frac{2440 \pm \sqrt{{(- 2440)}^{2} - 4 \cdot (- 351 \cdot 8 \cdot 10^{14})}}{2 \cdot - 351}$

Step 7

Simplify.

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Step 7.1

Multiply $- 351$ by $8$ .

$x = \frac{2440 \pm \sqrt{{(- 2440)}^{2} - 4 \cdot - 2808 \cdot 10^{14}}}{2 \cdot - 351}$

Step 7.2

Multiply $- 4$ by $- 2808$ .

$x = \frac{2440 \pm \sqrt{{(- 2440)}^{2} + 11232 \cdot 10^{14}}}{2 \cdot - 351}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Raise $- 2440$ to the power of $2$ .

$x = \frac{2440 \pm \sqrt{5953600 + 11232 \cdot 10^{14}}}{2 \cdot - 351}$

Step 7.3.2

Move the decimal point in $11232$ to the left by $4$ places and increase the power of $10^{14}$ by $4$ .

$x = \frac{2440 \pm \sqrt{5953600 + 1.1232 \cdot 10^{18}}}{2 \cdot - 351}$

Step 7.3.3

Convert $5953600$ to scientific notation.

$x = \frac{2440 \pm \sqrt{5.9536 \cdot 10^{6} + 1.1232 \cdot 10^{18}}}{2 \cdot - 351}$

Step 7.3.4

Move the decimal point in $5.9536$ to the left by $12$ places and increase the power of $10^{6}$ by $12$ .

$x = \frac{2440 \pm \sqrt{0 \cdot 10^{18} + 1.1232 \cdot 10^{18}}}{2 \cdot - 351}$

Step 7.3.5

Factor $10^{18}$ out of $0 \cdot 10^{18} + 1.1232 \cdot 10^{18}$ .

$x = \frac{2440 \pm \sqrt{(0 + 1.1232) \cdot 10^{18}}}{2 \cdot - 351}$

Step 7.3.6

Add $0$ and $1.1232$ .

$x = \frac{2440 \pm \sqrt{1.1232 \cdot 10^{18}}}{2 \cdot - 351}$

Step 7.3.7

Raise $10$ to the power of $18$ .

$x = \frac{2440 \pm \sqrt{1.1232 \cdot 1000000000000000000}}{2 \cdot - 351}$

Step 7.3.8

Multiply $1.1232$ by $1000000000000000000$ .

$x = \frac{2440 \pm \sqrt{1123200000005953600}}{2 \cdot - 351}$

Step 7.4

Multiply $2$ by $- 351$ .

$x = \frac{2440 \pm \sqrt{1123200000005953600}}{- 702}$

Step 7.5

Move the negative in front of the fraction.

$x = - \frac{2440 \pm \sqrt{1123200000005953600}}{702}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{2440 + \sqrt{1123200000005953600}}{702}, - \frac{2440 - \sqrt{1123200000005953600}}{702}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2440 + \sqrt{1123200000005953600}}{702}, - \frac{2440 - \sqrt{1123200000005953600}}{702}$

Decimal Form:

$x = - 1509706.18798027, 1509699.23641332$