Trigonometry Examples

$y = \frac{x}{x^{2} - 9}$

Step 1

Multiply the equation by $x^{2} - 9$ .

$(x^{2} - 9) (y) = (\frac{x}{x^{2} - 9}) (x^{2} - 9)$

Step 2

Simplify the left side.

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Step 2.1

Apply the distributive property.

$x^{2} y - 9 y = (\frac{x}{x^{2} - 9}) (x^{2} - 9)$

Step 3

Simplify the right side.

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Step 3.1

Simplify $(\frac{x}{x^{2} - 9}) (x^{2} - 9)$ .

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Step 3.1.1

Simplify the denominator.

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Step 3.1.1.1

Rewrite $9$ as $3^{2}$ .

$x^{2} y - 9 y = \frac{x}{x^{2} - 3^{2}} (x^{2} - 9)$

Step 3.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$x^{2} y - 9 y = \frac{x}{(x + 3) (x - 3)} (x^{2} - 9)$

Step 3.1.2

Multiply $\frac{x}{(x + 3) (x - 3)}$ by $x^{2} - 9$ .

$x^{2} y - 9 y = \frac{x (x^{2} - 9)}{(x + 3) (x - 3)}$

Step 3.1.3

Simplify the numerator.

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Step 3.1.3.1

Rewrite $9$ as $3^{2}$ .

$x^{2} y - 9 y = \frac{x (x^{2} - 3^{2})}{(x + 3) (x - 3)}$

Step 3.1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$x^{2} y - 9 y = \frac{x (x + 3) (x - 3)}{(x + 3) (x - 3)}$

Step 3.1.4

Reduce the expression by cancelling the common factors.

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Step 3.1.4.1

Cancel the common factor of $x + 3$ .

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Step 3.1.4.1.1

Cancel the common factor.

$x^{2} y - 9 y = \frac{x (x + 3) (x - 3)}{(x + 3) (x - 3)}$

Step 3.1.4.1.2

Rewrite the expression.

$x^{2} y - 9 y = \frac{x (x - 3)}{x - 3}$

Step 3.1.4.2

Cancel the common factor of $x - 3$ .

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Step 3.1.4.2.1

Cancel the common factor.

$x^{2} y - 9 y = \frac{x (x - 3)}{x - 3}$

Step 3.1.4.2.2

Divide $x$ by $1$ .

$x^{2} y - 9 y = x$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $x$ from both sides of the equation.

$x^{2} y - 9 y - x = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = y$ , $b = - 1$ , and $c = - 9 y$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (y \cdot (- 9 y))}}{2 y}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot y \cdot (- 9 y)}}{2 y}$

Step 4.4.2

Rewrite using the commutative property of multiplication.

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 9 y \cdot y)}}{2 y}$

Step 4.4.3

Multiply $y$ by $y$ by adding the exponents.

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Step 4.4.3.1

Move $y$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 9 (y \cdot y))}}{2 y}$

Step 4.4.3.2

Multiply $y$ by $y$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 9 y^{2})}}{2 y}$

Step 4.4.4

Multiply $- 4$ by $- 9$ .

$x = \frac{1 \pm \sqrt{1 + 36 y^{2}}}{2 y}$

Step 4.5

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{1 + 36 y^{2}}}{2 y}$

$x = \frac{1 - \sqrt{1 + 36 y^{2}}}{2 y}$

$x = \frac{1 + \sqrt{1 + 36 y^{2}}}{2 y}$

$x = \frac{1 - \sqrt{1 + 36 y^{2}}}{2 y}$