Trigonometry Examples

$\frac{13}{x} = 10 - x$

Step 1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

Step 1.2

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $\frac{13}{x} = 10 - x$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $\frac{13}{x} = 10 - x$ by $x$ .

$\frac{13}{x} x = 10 x - x \cdot x$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{13}{x} x = 10 x - x \cdot x$

Step 2.2.1.2

Rewrite the expression.

$13 = 10 x - x \cdot x$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Move $x$ .

$13 = 10 x - (x \cdot x)$

Step 2.3.1.2

Multiply $x$ by $x$ .

$13 = 10 x - x^{2}$

Step 3

Solve the equation.

Tap for more steps...

Step 3.1

Rewrite the equation as $10 x - x^{2} = 13$ .

$10 x - x^{2} = 13$

Step 3.2

Subtract $13$ from both sides of the equation.

$10 x - x^{2} - 13 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = - 1$ , $b = 10$ , and $c = - 13$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (- 1 \cdot - 13)}}{2 \cdot - 1}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Simplify the numerator.

Tap for more steps...

Step 3.5.1.1

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 13}}{2 \cdot - 1}$

Step 3.5.1.2

Multiply $- 4 \cdot - 1 \cdot - 13$ .

Tap for more steps...

Step 3.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 10 \pm \sqrt{100 + 4 \cdot - 13}}{2 \cdot - 1}$

Step 3.5.1.2.2

Multiply $4$ by $- 13$ .

$x = \frac{- 10 \pm \sqrt{100 - 52}}{2 \cdot - 1}$

Step 3.5.1.3

Subtract $52$ from $100$ .

$x = \frac{- 10 \pm \sqrt{48}}{2 \cdot - 1}$

Step 3.5.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 3.5.1.4.1

Factor $16$ out of $48$ .

$x = \frac{- 10 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

Step 3.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 10 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot - 1}$

Step 3.5.1.5

Pull terms out from under the radical.

$x = \frac{- 10 \pm 4 \sqrt{3}}{2 \cdot - 1}$

Step 3.5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 10 \pm 4 \sqrt{3}}{- 2}$

Step 3.5.3

Simplify $\frac{- 10 \pm 4 \sqrt{3}}{- 2}$ .

$x = 5 \pm 2 \sqrt{3}$

Step 3.6

The final answer is the combination of both solutions.

$x = 5 + 2 \sqrt{3}, 5 - 2 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 5 + 2 \sqrt{3}, 5 - 2 \sqrt{3}$

Decimal Form:

$x = 8.46410161 \dots, 1.53589838 \dots$