Trigonometry Examples

$x (18 - x) = 90$

Step 1

Simplify $x (18 - x)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x \cdot 18 + x (- x) = 90$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $18$ to the left of $x$ .

$18 \cdot x + x (- x) = 90$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$18 \cdot x - x \cdot x = 90$

Step 1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1

Move $x$ .

$18 x - (x \cdot x) = 90$

Step 1.2.2

Multiply $x$ by $x$ .

$18 x - x^{2} = 90$

Step 2

Subtract $90$ from both sides of the equation.

$18 x - x^{2} - 90 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 1$ , $b = 18$ , and $c = - 90$ into the quadratic formula and solve for $x$ .

$\frac{- 18 \pm \sqrt{18^{2} - 4 \cdot (- 1 \cdot - 90)}}{2 \cdot - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $18$ to the power of $2$ .

$x = \frac{- 18 \pm \sqrt{324 - 4 \cdot - 1 \cdot - 90}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot - 90$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 18 \pm \sqrt{324 + 4 \cdot - 90}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $- 90$ .

$x = \frac{- 18 \pm \sqrt{324 - 360}}{2 \cdot - 1}$

Step 5.1.3

Subtract $360$ from $324$ .

$x = \frac{- 18 \pm \sqrt{- 36}}{2 \cdot - 1}$

Step 5.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 18 \pm \sqrt{- 1 \cdot 36}}{2 \cdot - 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{- 18 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot - 1}$

Step 5.1.6

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

$x = \frac{- 18 \pm i \cdot \sqrt{36}}{2 \cdot - 1}$

Step 5.1.7

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 18 \pm i \cdot \sqrt{6^{2}}}{2 \cdot - 1}$

Step 5.1.8

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

$x = \frac{- 18 \pm i \cdot 6}{2 \cdot - 1}$

Step 5.1.9

Move $6$ to the left of $i$ .

$x = \frac{- 18 \pm 6 i}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 18 \pm 6 i}{- 2}$

Step 5.3

Simplify $\frac{- 18 \pm 6 i}{- 2}$ .

$x = 9 \pm 3 i$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $18$ to the power of $2$ .

$x = \frac{- 18 \pm \sqrt{324 - 4 \cdot - 1 \cdot - 90}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot - 90$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 18 \pm \sqrt{324 + 4 \cdot - 90}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $- 90$ .

$x = \frac{- 18 \pm \sqrt{324 - 360}}{2 \cdot - 1}$

Step 6.1.3

Subtract $360$ from $324$ .

$x = \frac{- 18 \pm \sqrt{- 36}}{2 \cdot - 1}$

Step 6.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 18 \pm \sqrt{- 1 \cdot 36}}{2 \cdot - 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{- 18 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot - 1}$

Step 6.1.6

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

$x = \frac{- 18 \pm i \cdot \sqrt{36}}{2 \cdot - 1}$

Step 6.1.7

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 18 \pm i \cdot \sqrt{6^{2}}}{2 \cdot - 1}$

Step 6.1.8

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

$x = \frac{- 18 \pm i \cdot 6}{2 \cdot - 1}$

Step 6.1.9

Move $6$ to the left of $i$ .

$x = \frac{- 18 \pm 6 i}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 18 \pm 6 i}{- 2}$

Step 6.3

Simplify $\frac{- 18 \pm 6 i}{- 2}$ .

$x = 9 \pm 3 i$

Step 6.4

Change the $\pm$ to $+$ .

$x = 9 + 3 i$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $18$ to the power of $2$ .

$x = \frac{- 18 \pm \sqrt{324 - 4 \cdot - 1 \cdot - 90}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 90$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 18 \pm \sqrt{324 + 4 \cdot - 90}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 90$ .

$x = \frac{- 18 \pm \sqrt{324 - 360}}{2 \cdot - 1}$

Step 7.1.3

Subtract $360$ from $324$ .

$x = \frac{- 18 \pm \sqrt{- 36}}{2 \cdot - 1}$

Step 7.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 18 \pm \sqrt{- 1 \cdot 36}}{2 \cdot - 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{- 18 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot - 1}$

Step 7.1.6

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

$x = \frac{- 18 \pm i \cdot \sqrt{36}}{2 \cdot - 1}$

Step 7.1.7

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 18 \pm i \cdot \sqrt{6^{2}}}{2 \cdot - 1}$

Step 7.1.8

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

$x = \frac{- 18 \pm i \cdot 6}{2 \cdot - 1}$

Step 7.1.9

Move $6$ to the left of $i$ .

$x = \frac{- 18 \pm 6 i}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 18 \pm 6 i}{- 2}$

Step 7.3

Simplify $\frac{- 18 \pm 6 i}{- 2}$ .

$x = 9 \pm 3 i$

Step 7.4

Change the $\pm$ to $-$ .

$x = 9 - 3 i$

Step 8

The final answer is the combination of both solutions.

$x = 9 + 3 i, 9 - 3 i$