Trigonometry Examples

$6 n^{2} + 6 = 7 n$

Step 1

Subtract $7 n$ from both sides of the equation.

$6 n^{2} + 6 - 7 n = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 6$ , $b = - 7$ , and $c = 6$ into the quadratic formula and solve for $n$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (6 \cdot 6)}}{2 \cdot 6}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 7$ to the power of $2$ .

$n = \frac{7 \pm \sqrt{49 - 4 \cdot 6 \cdot 6}}{2 \cdot 6}$

Step 4.1.2

Multiply $- 4 \cdot 6 \cdot 6$ .

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Step 4.1.2.1

Multiply $- 4$ by $6$ .

$n = \frac{7 \pm \sqrt{49 - 24 \cdot 6}}{2 \cdot 6}$

Step 4.1.2.2

Multiply $- 24$ by $6$ .

$n = \frac{7 \pm \sqrt{49 - 144}}{2 \cdot 6}$

Step 4.1.3

Subtract $144$ from $49$ .

$n = \frac{7 \pm \sqrt{- 95}}{2 \cdot 6}$

Step 4.1.4

Rewrite $- 95$ as $- 1 (95)$ .

$n = \frac{7 \pm \sqrt{- 1 \cdot 95}}{2 \cdot 6}$

Step 4.1.5

Rewrite $\sqrt{- 1 (95)}$ as $\sqrt{- 1} \cdot \sqrt{95}$ .

$n = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{95}}{2 \cdot 6}$

Step 4.1.6

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

$n = \frac{7 \pm i \sqrt{95}}{2 \cdot 6}$

Step 4.2

Multiply $2$ by $6$ .

$n = \frac{7 \pm i \sqrt{95}}{12}$

Step 5

The final answer is the combination of both solutions.

$n = \frac{7 + i \sqrt{95}}{12}, \frac{7 - i \sqrt{95}}{12}$