Trigonometry Examples

$8 x^{2} - 5 x - 1 = 0$

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 $a = 8$ , $b = - 5$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (8 \cdot - 1)}}{2 \cdot 8}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 8 \cdot - 1}}{2 \cdot 8}$

Step 3.1.2

Multiply $- 4 \cdot 8 \cdot - 1$ .

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Step 3.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{5 \pm \sqrt{25 - 32 \cdot - 1}}{2 \cdot 8}$

Step 3.1.2.2

Multiply $- 32$ by $- 1$ .

$x = \frac{5 \pm \sqrt{25 + 32}}{2 \cdot 8}$

Step 3.1.3

Add $25$ and $32$ .

$x = \frac{5 \pm \sqrt{57}}{2 \cdot 8}$

Step 3.2

Multiply $2$ by $8$ .

$x = \frac{5 \pm \sqrt{57}}{16}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{57}}{16}, \frac{5 - \sqrt{57}}{16}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{57}}{16}, \frac{5 - \sqrt{57}}{16}$

Decimal Form:

$x = 0.78436465 \dots, - 0.15936465 \dots$