Finite Math Examples

$10 x^{2} + 23 x + 1 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 10$ , $b = 23$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 23 \pm \sqrt{23^{2} - 4 \cdot (10 \cdot 1)}}{2 \cdot 10}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot 10 \cdot 1}}{2 \cdot 10}$

Step 3.1.2

Multiply $- 4 \cdot 10 \cdot 1$ .

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Step 3.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{- 23 \pm \sqrt{529 - 40 \cdot 1}}{2 \cdot 10}$

Step 3.1.2.2

Multiply $- 40$ by $1$ .

$x = \frac{- 23 \pm \sqrt{529 - 40}}{2 \cdot 10}$

Step 3.1.3

Subtract $40$ from $529$ .

$x = \frac{- 23 \pm \sqrt{489}}{2 \cdot 10}$

Step 3.2

Multiply $2$ by $10$ .

$x = \frac{- 23 \pm \sqrt{489}}{20}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot 10 \cdot 1}}{2 \cdot 10}$

Step 4.1.2

Multiply $- 4 \cdot 10 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{- 23 \pm \sqrt{529 - 40 \cdot 1}}{2 \cdot 10}$

Step 4.1.2.2

Multiply $- 40$ by $1$ .

$x = \frac{- 23 \pm \sqrt{529 - 40}}{2 \cdot 10}$

Step 4.1.3

Subtract $40$ from $529$ .

$x = \frac{- 23 \pm \sqrt{489}}{2 \cdot 10}$

Step 4.2

Multiply $2$ by $10$ .

$x = \frac{- 23 \pm \sqrt{489}}{20}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{- 23 + \sqrt{489}}{20}$

Step 4.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 1 \cdot 23 + \sqrt{489}}{20}$

Step 4.5

Factor $- 1$ out of $\sqrt{489}$ .

$x = \frac{- 1 \cdot 23 - 1 (- \sqrt{489})}{20}$

Step 4.6

Factor $- 1$ out of $- 1 (23) - 1 (- \sqrt{489})$ .

$x = \frac{- 1 (23 - \sqrt{489})}{20}$

Step 4.7

Move the negative in front of the fraction.

$x = - \frac{23 - \sqrt{489}}{20}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot 10 \cdot 1}}{2 \cdot 10}$

Step 5.1.2

Multiply $- 4 \cdot 10 \cdot 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{- 23 \pm \sqrt{529 - 40 \cdot 1}}{2 \cdot 10}$

Step 5.1.2.2

Multiply $- 40$ by $1$ .

$x = \frac{- 23 \pm \sqrt{529 - 40}}{2 \cdot 10}$

Step 5.1.3

Subtract $40$ from $529$ .

$x = \frac{- 23 \pm \sqrt{489}}{2 \cdot 10}$

Step 5.2

Multiply $2$ by $10$ .

$x = \frac{- 23 \pm \sqrt{489}}{20}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{- 23 - \sqrt{489}}{20}$

Step 5.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 1 \cdot 23 - \sqrt{489}}{20}$

Step 5.5

Factor $- 1$ out of $- \sqrt{489}$ .

$x = \frac{- 1 \cdot 23 - (\sqrt{489})}{20}$

Step 5.6

Factor $- 1$ out of $- 1 (23) - (\sqrt{489})$ .

$x = \frac{- 1 (23 + \sqrt{489})}{20}$

Step 5.7

Move the negative in front of the fraction.

$x = - \frac{23 + \sqrt{489}}{20}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{23 - \sqrt{489}}{20}, - \frac{23 + \sqrt{489}}{20}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{23 - \sqrt{489}}{20}, - \frac{23 + \sqrt{489}}{20}$

Decimal Form:

$x = - 0.04433278 \dots, - 2.25566721 \dots$