Finite Math Examples

$0.149 x^{2} + 0.014 x - 0.408 = 0$

Step 1

Factor $0.001$ out of $0.149 x^{2} + 0.014 x - 0.408$ .

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Step 1.1

Factor $0.001$ out of $0.149 x^{2}$ .

$0.001 (149 x^{2}) + 0.014 x - 0.408 = 0$

Step 1.2

Factor $0.001$ out of $0.014 x$ .

$0.001 (149 x^{2}) + 0.001 (14 x) - 0.408 = 0$

Step 1.3

Factor $0.001$ out of $- 0.408$ .

$0.001 (149 x^{2}) + 0.001 (14 x) + 0.001 (- 408) = 0$

Step 1.4

Factor $0.001$ out of $0.001 (149 x^{2}) + 0.001 (14 x)$ .

$0.001 (149 x^{2} + 14 x) + 0.001 (- 408) = 0$

Step 1.5

Factor $0.001$ out of $0.001 (149 x^{2} + 14 x) + 0.001 (- 408)$ .

$0.001 (149 x^{2} + 14 x - 408) = 0$

Step 2

Divide each term in $0.001 (149 x^{2} + 14 x - 408) = 0$ by $0.001$ and simplify.

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Step 2.1

Divide each term in $0.001 (149 x^{2} + 14 x - 408) = 0$ by $0.001$ .

$\frac{0.001 (149 x^{2} + 14 x - 408)}{0.001} = \frac{0}{0.001}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $0.001$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{0.001 (149 x^{2} + 14 x - 408)}{0.001} = \frac{0}{0.001}$

Step 2.2.1.2

Divide $149 x^{2} + 14 x - 408$ by $1$ .

$149 x^{2} + 14 x - 408 = \frac{0}{0.001}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $0.001$ .

$149 x^{2} + 14 x - 408 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 149$ , $b = 14$ , and $c = - 408$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (149 \cdot - 408)}}{2 \cdot 149}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 149 \cdot - 408}}{2 \cdot 149}$

Step 5.1.2

Multiply $- 4 \cdot 149 \cdot - 408$ .

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Step 5.1.2.1

Multiply $- 4$ by $149$ .

$x = \frac{- 14 \pm \sqrt{196 - 596 \cdot - 408}}{2 \cdot 149}$

Step 5.1.2.2

Multiply $- 596$ by $- 408$ .

$x = \frac{- 14 \pm \sqrt{196 + 243168}}{2 \cdot 149}$

Step 5.1.3

Add $196$ and $243168$ .

$x = \frac{- 14 \pm \sqrt{243364}}{2 \cdot 149}$

Step 5.1.4

Rewrite $243364$ as $2^{2} \cdot 60841$ .

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Step 5.1.4.1

Factor $4$ out of $243364$ .

$x = \frac{- 14 \pm \sqrt{4 (60841)}}{2 \cdot 149}$

Step 5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 60841}}{2 \cdot 149}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{60841}}{2 \cdot 149}$

Step 5.2

Multiply $2$ by $149$ .

$x = \frac{- 14 \pm 2 \sqrt{60841}}{298}$

Step 5.3

Simplify $\frac{- 14 \pm 2 \sqrt{60841}}{298}$ .

$x = \frac{- 7 \pm \sqrt{60841}}{149}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 149 \cdot - 408}}{2 \cdot 149}$

Step 6.1.2

Multiply $- 4 \cdot 149 \cdot - 408$ .

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Step 6.1.2.1

Multiply $- 4$ by $149$ .

$x = \frac{- 14 \pm \sqrt{196 - 596 \cdot - 408}}{2 \cdot 149}$

Step 6.1.2.2

Multiply $- 596$ by $- 408$ .

$x = \frac{- 14 \pm \sqrt{196 + 243168}}{2 \cdot 149}$

Step 6.1.3

Add $196$ and $243168$ .

$x = \frac{- 14 \pm \sqrt{243364}}{2 \cdot 149}$

Step 6.1.4

Rewrite $243364$ as $2^{2} \cdot 60841$ .

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Step 6.1.4.1

Factor $4$ out of $243364$ .

$x = \frac{- 14 \pm \sqrt{4 (60841)}}{2 \cdot 149}$

Step 6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 60841}}{2 \cdot 149}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{60841}}{2 \cdot 149}$

Step 6.2

Multiply $2$ by $149$ .

$x = \frac{- 14 \pm 2 \sqrt{60841}}{298}$

Step 6.3

Simplify $\frac{- 14 \pm 2 \sqrt{60841}}{298}$ .

$x = \frac{- 7 \pm \sqrt{60841}}{149}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{60841}}{149}$

Step 6.5

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{60841}}{149}$

Step 6.6

Factor $- 1$ out of $\sqrt{60841}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{60841})}{149}$

Step 6.7

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{60841})$ .

$x = \frac{- 1 (7 - \sqrt{60841})}{149}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{60841}}{149}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 149 \cdot - 408}}{2 \cdot 149}$

Step 7.1.2

Multiply $- 4 \cdot 149 \cdot - 408$ .

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Step 7.1.2.1

Multiply $- 4$ by $149$ .

$x = \frac{- 14 \pm \sqrt{196 - 596 \cdot - 408}}{2 \cdot 149}$

Step 7.1.2.2

Multiply $- 596$ by $- 408$ .

$x = \frac{- 14 \pm \sqrt{196 + 243168}}{2 \cdot 149}$

Step 7.1.3

Add $196$ and $243168$ .

$x = \frac{- 14 \pm \sqrt{243364}}{2 \cdot 149}$

Step 7.1.4

Rewrite $243364$ as $2^{2} \cdot 60841$ .

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Step 7.1.4.1

Factor $4$ out of $243364$ .

$x = \frac{- 14 \pm \sqrt{4 (60841)}}{2 \cdot 149}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 60841}}{2 \cdot 149}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{60841}}{2 \cdot 149}$

Step 7.2

Multiply $2$ by $149$ .

$x = \frac{- 14 \pm 2 \sqrt{60841}}{298}$

Step 7.3

Simplify $\frac{- 14 \pm 2 \sqrt{60841}}{298}$ .

$x = \frac{- 7 \pm \sqrt{60841}}{149}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{60841}}{149}$

Step 7.5

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{60841}}{149}$

Step 7.6

Factor $- 1$ out of $- \sqrt{60841}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{60841})}{149}$

Step 7.7

Factor $- 1$ out of $- 1 (7) - (\sqrt{60841})$ .

$x = \frac{- 1 (7 + \sqrt{60841})}{149}$

Step 7.8

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{60841}}{149}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{60841}}{149}, - \frac{7 + \sqrt{60841}}{149}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 - \sqrt{60841}}{149}, - \frac{7 + \sqrt{60841}}{149}$

Decimal Form:

$x = 1.60845425 \dots, - 1.70241399 \dots$