Calculus Examples

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

Step 1

Rewrite the equation as $\frac{x}{x^{2} + 9} = y$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$x^{2} + 9, 1$

Step 2.2

Remove parentheses.

$x^{2} + 9, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x^{2} + 9$

Step 3

Multiply each term in $\frac{x}{x^{2} + 9} = y$ by $x^{2} + 9$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{x}{x^{2} + 9} = y$ by $x^{2} + 9$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{2} + 9$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

$x = y (x^{2} + 9)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$x = y x^{2} + y \cdot 9$

Step 3.3.2

Move $9$ to the left of $y$ .

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

Step 4

Solve the equation.

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Step 4.1

Subtract $y x^{2}$ from both sides of the equation.

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

Step 4.2

Subtract $9 y$ from both sides of the equation.

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

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

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 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Rewrite $- 4 y \cdot (- 9 y)$ as ${(6 y)}^{2}$ .

$x = \frac{- 1 \pm \sqrt{1^{2} - {(6 y)}^{2}}}{2 (- y)}$

Step 4.5.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 6 y$ .

$x = \frac{- 1 \pm \sqrt{(1 + 6 y) (1 - (6 y))}}{2 (- y)}$

Step 4.5.1.3

Multiply $6$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{(1 + 6 y) (1 - 6 y)}}{2 (- y)}$

Step 4.5.2

Multiply $- 1$ by $2$ .

$x = \frac{- 1 \pm \sqrt{(1 + 6 y) (1 - 6 y)}}{- 2 y}$

Step 4.5.3

Simplify $\frac{- 1 \pm \sqrt{(1 + 6 y) (1 - 6 y)}}{- 2 y}$ .

$x = \frac{1 \pm \sqrt{(1 + 6 y) (1 - 6 y)}}{2 y}$

Step 4.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{(1 + 6 y) (1 - 6 y)}}{2 y}$

$x = \frac{1 - \sqrt{(1 + 6 y) (1 - 6 y)}}{2 y}$

$x = \frac{1 + \sqrt{(1 + 6 y) (1 - 6 y)}}{2 y}$

$x = \frac{1 - \sqrt{(1 + 6 y) (1 - 6 y)}}{2 y}$