Finite Math Examples

$x^{2} - 3 x - 1 = 0$

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 $a = 1$ , $b = - 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 3.1.2

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

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 3.1.3

Add $9$ and $4$ .

$x = \frac{3 \pm \sqrt{13}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{13}}{2}$

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 $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 4.1.3

Add $9$ and $4$ .

$x = \frac{3 \pm \sqrt{13}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{13}}{2}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{13}}{2}$

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 $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 5.1.3

Add $9$ and $4$ .

$x = \frac{3 \pm \sqrt{13}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{13}}{2}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{13}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{13}}{2}, \frac{3 - \sqrt{13}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{13}}{2}, \frac{3 - \sqrt{13}}{2}$

Decimal Form:

$x = 3.30277563 \dots, - 0.30277563 \dots$