Trigonometry Examples

$x^{2} - 15 x + 56.5 = 0$

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 $a = 1$ , $b = - 15$ , and $c = 56.5$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (1 \cdot 56.5)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 56.5}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 56.5$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 56.5}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $56.5$ .

$x = \frac{15 \pm \sqrt{225 - 226}}{2 \cdot 1}$

Step 3.1.3

Subtract $226$ from $225$ .

$x = \frac{15 \pm \sqrt{- 1}}{2 \cdot 1}$

Step 3.1.4

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

$x = \frac{15 \pm \sqrt{- 1 \cdot 1}}{2 \cdot 1}$

Step 3.1.5

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

$x = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{1}}{2 \cdot 1}$

Step 3.1.6

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

$x = \frac{15 \pm i \cdot \sqrt{1}}{2 \cdot 1}$

Step 3.1.7

Rewrite $1$ as $1^{2}$ .

$x = \frac{15 \pm i \cdot \sqrt{1^{2}}}{2 \cdot 1}$

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{15 \pm i \cdot 1}{2 \cdot 1}$

Step 3.1.9

Multiply $i$ by $1$ .

$x = \frac{15 \pm i}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{15 \pm i}{2}$

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 $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 56.5}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 56.5$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 56.5}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $56.5$ .

$x = \frac{15 \pm \sqrt{225 - 226}}{2 \cdot 1}$

Step 4.1.3

Subtract $226$ from $225$ .

$x = \frac{15 \pm \sqrt{- 1}}{2 \cdot 1}$

Step 4.1.4

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

$x = \frac{15 \pm \sqrt{- 1 \cdot 1}}{2 \cdot 1}$

Step 4.1.5

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

$x = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{1}}{2 \cdot 1}$

Step 4.1.6

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

$x = \frac{15 \pm i \cdot \sqrt{1}}{2 \cdot 1}$

Step 4.1.7

Rewrite $1$ as $1^{2}$ .

$x = \frac{15 \pm i \cdot \sqrt{1^{2}}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{15 \pm i \cdot 1}{2 \cdot 1}$

Step 4.1.9

Multiply $i$ by $1$ .

$x = \frac{15 \pm i}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{15 \pm i}{2}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{15 + i}{2}$

Step 4.4

Split the fraction $\frac{15 + i}{2}$ into two fractions.

$x = \frac{15}{2} + \frac{i}{2}$

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 $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 56.5}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 56.5$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 56.5}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $56.5$ .

$x = \frac{15 \pm \sqrt{225 - 226}}{2 \cdot 1}$

Step 5.1.3

Subtract $226$ from $225$ .

$x = \frac{15 \pm \sqrt{- 1}}{2 \cdot 1}$

Step 5.1.4

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

$x = \frac{15 \pm \sqrt{- 1 \cdot 1}}{2 \cdot 1}$

Step 5.1.5

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

$x = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{1}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{15 \pm i \cdot \sqrt{1}}{2 \cdot 1}$

Step 5.1.7

Rewrite $1$ as $1^{2}$ .

$x = \frac{15 \pm i \cdot \sqrt{1^{2}}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{15 \pm i \cdot 1}{2 \cdot 1}$

Step 5.1.9

Multiply $i$ by $1$ .

$x = \frac{15 \pm i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{15 \pm i}{2}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{15 - i}{2}$

Step 5.4

Split the fraction $\frac{15 - i}{2}$ into two fractions.

$x = \frac{15}{2} + \frac{- i}{2}$

Step 5.5

Move the negative in front of the fraction.

$x = \frac{15}{2} - \frac{i}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{15}{2} + \frac{i}{2}, \frac{15}{2} - \frac{i}{2}$