Finite Math Examples

$2 x^{2} + 8 x - 1 = 0$

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 $a = 2$ , $b = 8$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 3.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 8 \cdot - 1}}{2 \cdot 2}$

Step 3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{- 8 \pm \sqrt{64 + 8}}{2 \cdot 2}$

Step 3.1.3

Add $64$ and $8$ .

$x = \frac{- 8 \pm \sqrt{72}}{2 \cdot 2}$

Step 3.1.4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 3.1.4.1

Factor $36$ out of $72$ .

$x = \frac{- 8 \pm \sqrt{36 (2)}}{2 \cdot 2}$

Step 3.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 8 \pm \sqrt{6^{2} \cdot 2}}{2 \cdot 2}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 6 \sqrt{2}}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$x = \frac{- 8 \pm 6 \sqrt{2}}{4}$

Step 3.3

Simplify $\frac{- 8 \pm 6 \sqrt{2}}{4}$ .

$x = \frac{- 4 \pm 3 \sqrt{2}}{2}$

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 $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 8 \cdot - 1}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{- 8 \pm \sqrt{64 + 8}}{2 \cdot 2}$

Step 4.1.3

Add $64$ and $8$ .

$x = \frac{- 8 \pm \sqrt{72}}{2 \cdot 2}$

Step 4.1.4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 4.1.4.1

Factor $36$ out of $72$ .

$x = \frac{- 8 \pm \sqrt{36 (2)}}{2 \cdot 2}$

Step 4.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 8 \pm \sqrt{6^{2} \cdot 2}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 6 \sqrt{2}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 8 \pm 6 \sqrt{2}}{4}$

Step 4.3

Simplify $\frac{- 8 \pm 6 \sqrt{2}}{4}$ .

$x = \frac{- 4 \pm 3 \sqrt{2}}{2}$

Step 4.4

Change the $\pm$ to $+$ .

$x = \frac{- 4 + 3 \sqrt{2}}{2}$

Step 4.5

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 1 \cdot 4 + 3 \sqrt{2}}{2}$

Step 4.6

Factor $- 1$ out of $3 \sqrt{2}$ .

$x = \frac{- 1 \cdot 4 - (- 3 \sqrt{2})}{2}$

Step 4.7

Factor $- 1$ out of $- 1 (4) - (- 3 \sqrt{2})$ .

$x = \frac{- 1 (4 - 3 \sqrt{2})}{2}$

Step 4.8

Move the negative in front of the fraction.

$x = - \frac{4 - 3 \sqrt{2}}{2}$

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 $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 8 \cdot - 1}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{- 8 \pm \sqrt{64 + 8}}{2 \cdot 2}$

Step 5.1.3

Add $64$ and $8$ .

$x = \frac{- 8 \pm \sqrt{72}}{2 \cdot 2}$

Step 5.1.4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 5.1.4.1

Factor $36$ out of $72$ .

$x = \frac{- 8 \pm \sqrt{36 (2)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 8 \pm \sqrt{6^{2} \cdot 2}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 6 \sqrt{2}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 8 \pm 6 \sqrt{2}}{4}$

Step 5.3

Simplify $\frac{- 8 \pm 6 \sqrt{2}}{4}$ .

$x = \frac{- 4 \pm 3 \sqrt{2}}{2}$

Step 5.4

Change the $\pm$ to $-$ .

$x = \frac{- 4 - 3 \sqrt{2}}{2}$

Step 5.5

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 1 \cdot 4 - 3 \sqrt{2}}{2}$

Step 5.6

Factor $- 1$ out of $- 3 \sqrt{2}$ .

$x = \frac{- 1 \cdot 4 - (3 \sqrt{2})}{2}$

Step 5.7

Factor $- 1$ out of $- 1 (4) - (3 \sqrt{2})$ .

$x = \frac{- 1 (4 + 3 \sqrt{2})}{2}$

Step 5.8

Move the negative in front of the fraction.

$x = - \frac{4 + 3 \sqrt{2}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{4 - 3 \sqrt{2}}{2}, - \frac{4 + 3 \sqrt{2}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4 - 3 \sqrt{2}}{2}, - \frac{4 + 3 \sqrt{2}}{2}$

Decimal Form:

$x = 0.12132034 \dots, - 4.12132034 \dots$