Trigonometry Examples

$4 x^{2} - 3 x - 8 = 0$

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 $a = 4$ , $b = - 3$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (4 \cdot - 8)}}{2 \cdot 4}$

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 $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 8}}{2 \cdot 4}$

Step 3.1.2

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

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Step 3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 8}}{2 \cdot 4}$

Step 3.1.2.2

Multiply $- 16$ by $- 8$ .

$x = \frac{3 \pm \sqrt{9 + 128}}{2 \cdot 4}$

Step 3.1.3

Add $9$ and $128$ .

$x = \frac{3 \pm \sqrt{137}}{2 \cdot 4}$

Step 3.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{137}}{8}$

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 $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 8}}{2 \cdot 4}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 8}}{2 \cdot 4}$

Step 4.1.2.2

Multiply $- 16$ by $- 8$ .

$x = \frac{3 \pm \sqrt{9 + 128}}{2 \cdot 4}$

Step 4.1.3

Add $9$ and $128$ .

$x = \frac{3 \pm \sqrt{137}}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{137}}{8}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{137}}{8}$

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 $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 8}}{2 \cdot 4}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 8}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 8$ .

$x = \frac{3 \pm \sqrt{9 + 128}}{2 \cdot 4}$

Step 5.1.3

Add $9$ and $128$ .

$x = \frac{3 \pm \sqrt{137}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{137}}{8}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{137}}{8}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{137}}{8}, \frac{3 - \sqrt{137}}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{137}}{8}, \frac{3 - \sqrt{137}}{8}$

Decimal Form:

$x = 1.83808748 \dots, - 1.08808748 \dots$