Trigonometry Examples

$\frac{5}{x} = \frac{x + 3}{6}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$5 \cdot 6 = x (x + 3)$

Step 2

Solve the equation for $x$ .

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Step 2.1

Rewrite the equation as $x (x + 3) = 5 \cdot 6$ .

$x (x + 3) = 5 \cdot 6$

Step 2.2

Simplify $x (x + 3)$ .

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Step 2.2.1

Apply the distributive property.

$x \cdot x + x \cdot 3 = 5 \cdot 6$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 = 5 \cdot 6$

Step 2.2.2.2

Move $3$ to the left of $x$ .

$x^{2} + 3 x = 5 \cdot 6$

Step 2.3

Multiply $5$ by $6$ .

$x^{2} + 3 x = 30$

Step 2.4

Subtract $30$ from both sides of the equation.

$x^{2} + 3 x - 30 = 0$

Step 2.5

Use the quadratic formula to find the solutions.

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

Step 2.6

Substitute the values $a = 1$ , $b = 3$ , and $c = - 30$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 30)}}{2 \cdot 1}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 30}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 30$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 30}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $- 30$ .

$x = \frac{- 3 \pm \sqrt{9 + 120}}{2 \cdot 1}$

Step 2.7.1.3

Add $9$ and $120$ .

$x = \frac{- 3 \pm \sqrt{129}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{129}}{2}$

Step 2.8

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{129}}{2}, - \frac{3 + \sqrt{129}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{129}}{2}, - \frac{3 + \sqrt{129}}{2}$

Decimal Form:

$x = 4.17890834 \dots, - 7.17890834 \dots$