Trigonometry Examples

$12 x^{2} - 5 x + 1 = 0$

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 $a = 12$ , $b = - 5$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (12 \cdot 1)}}{2 \cdot 12}$

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 $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 12 \cdot 1}}{2 \cdot 12}$

Step 3.1.2

Multiply $- 4 \cdot 12 \cdot 1$ .

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Step 3.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{5 \pm \sqrt{25 - 48 \cdot 1}}{2 \cdot 12}$

Step 3.1.2.2

Multiply $- 48$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 48}}{2 \cdot 12}$

Step 3.1.3

Subtract $48$ from $25$ .

$x = \frac{5 \pm \sqrt{- 23}}{2 \cdot 12}$

Step 3.1.4

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

$x = \frac{5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 12}$

Step 3.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 12}$

Step 3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{23}}{2 \cdot 12}$

Step 3.2

Multiply $2$ by $12$ .

$x = \frac{5 \pm i \sqrt{23}}{24}$

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 12 \cdot 1}}{2 \cdot 12}$

Step 4.1.2

Multiply $- 4 \cdot 12 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{5 \pm \sqrt{25 - 48 \cdot 1}}{2 \cdot 12}$

Step 4.1.2.2

Multiply $- 48$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 48}}{2 \cdot 12}$

Step 4.1.3

Subtract $48$ from $25$ .

$x = \frac{5 \pm \sqrt{- 23}}{2 \cdot 12}$

Step 4.1.4

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

$x = \frac{5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 12}$

Step 4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 12}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{23}}{2 \cdot 12}$

Step 4.2

Multiply $2$ by $12$ .

$x = \frac{5 \pm i \sqrt{23}}{24}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{23}}{24}$

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 12 \cdot 1}}{2 \cdot 12}$

Step 5.1.2

Multiply $- 4 \cdot 12 \cdot 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{5 \pm \sqrt{25 - 48 \cdot 1}}{2 \cdot 12}$

Step 5.1.2.2

Multiply $- 48$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 48}}{2 \cdot 12}$

Step 5.1.3

Subtract $48$ from $25$ .

$x = \frac{5 \pm \sqrt{- 23}}{2 \cdot 12}$

Step 5.1.4

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

$x = \frac{5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 12}$

Step 5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 12}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{23}}{2 \cdot 12}$

Step 5.2

Multiply $2$ by $12$ .

$x = \frac{5 \pm i \sqrt{23}}{24}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - i \sqrt{23}}{24}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{23}}{24}, \frac{5 - i \sqrt{23}}{24}$