Precalculus Examples

$\frac{x^{3} - y^{3}}{x + y} \cdot \frac{x^{2} - y^{2}}{x^{2} + x y + y^{2}}$

Step 1

Set the denominator in $\frac{x^{3} - y^{3}}{x + y}$ equal to $0$ to find where the expression is undefined.

$x + y = 0$

Step 2

Subtract $y$ from both sides of the equation.

$x = - y$

Step 3

Set the denominator in $\frac{x^{2} - y^{2}}{x^{2} + x y + y^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x y + y^{2} = 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Use the quadratic formula to find the solutions.

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

Step 4.2

Substitute the values $a = 1$ , $b = y$ , and $c = y^{2}$ into the quadratic formula and solve for $x$ .

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

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Simplify the numerator.

Tap for more steps...

Step 4.3.1.1

Factor $y^{2}$ out of $y^{2} - 4 \cdot 1 \cdot y^{2}$ .

Tap for more steps...

Step 4.3.1.1.1

Multiply by $1$ .

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

Step 4.3.1.1.2

Factor $y^{2}$ out of $- 4 \cdot 1 \cdot y^{2}$ .

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

Step 4.3.1.1.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} \cdot (- 4 \cdot 1)$ .

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

Step 4.3.1.2

Multiply $- 4$ by $1$ .

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

Step 4.3.1.3

Subtract $4$ from $1$ .

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

Step 4.3.1.4

Pull terms out from under the radical.

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

Step 4.3.1.5

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.3.1.6

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 4.3.1.7

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

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

Step 4.3.2

Multiply $2$ by $1$ .

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

Step 4.4

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

Tap for more steps...

Step 4.4.1

Simplify the numerator.

Tap for more steps...

Step 4.4.1.1

Factor $y^{2}$ out of $y^{2} - 4 \cdot 1 \cdot y^{2}$ .

Tap for more steps...

Step 4.4.1.1.1

Multiply by $1$ .

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

Step 4.4.1.1.2

Factor $y^{2}$ out of $- 4 \cdot 1 \cdot y^{2}$ .

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

Step 4.4.1.1.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} \cdot (- 4 \cdot 1)$ .

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

Step 4.4.1.2

Multiply $- 4$ by $1$ .

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

Step 4.4.1.3

Subtract $4$ from $1$ .

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

Step 4.4.1.4

Pull terms out from under the radical.

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

Step 4.4.1.5

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.4.1.6

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 4.4.1.7

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

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

Step 4.4.2

Multiply $2$ by $1$ .

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

Step 4.4.3

Change the $\pm$ to $+$ .

$x = \frac{- y + y i \sqrt{3}}{2}$

Step 4.4.4

Factor $y$ out of $- y + y i \sqrt{3}$ .

Tap for more steps...

Step 4.4.4.1

Factor $y$ out of $- y$ .

$x = \frac{y \cdot - 1 + y i \sqrt{3}}{2}$

Step 4.4.4.2

Factor $y$ out of $y i \sqrt{3}$ .

$x = \frac{y \cdot - 1 + y (i \sqrt{3})}{2}$

Step 4.4.4.3

Factor $y$ out of $y \cdot - 1 + y (i \sqrt{3})$ .

$x = \frac{y (- 1 + i \sqrt{3})}{2}$

Step 4.4.5

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

$x = \frac{y (- 1 \cdot 1 + i \sqrt{3})}{2}$

Step 4.4.6

Factor $- 1$ out of $i \sqrt{3}$ .

$x = \frac{y (- 1 \cdot 1 - (- i \sqrt{3}))}{2}$

Step 4.4.7

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

$x = \frac{y (- 1 (1 - i \sqrt{3}))}{2}$

Step 4.4.8

Move the negative in front of the fraction.

$x = - \frac{y (1 - i \sqrt{3})}{2}$

Step 4.5

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

Tap for more steps...

Step 4.5.1

Simplify the numerator.

Tap for more steps...

Step 4.5.1.1

Factor $y^{2}$ out of $y^{2} - 4 \cdot 1 \cdot y^{2}$ .

Tap for more steps...

Step 4.5.1.1.1

Multiply by $1$ .

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

Step 4.5.1.1.2

Factor $y^{2}$ out of $- 4 \cdot 1 \cdot y^{2}$ .

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

Step 4.5.1.1.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} \cdot (- 4 \cdot 1)$ .

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

Step 4.5.1.2

Multiply $- 4$ by $1$ .

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

Step 4.5.1.3

Subtract $4$ from $1$ .

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

Step 4.5.1.4

Pull terms out from under the radical.

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

Step 4.5.1.5

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.5.1.6

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 4.5.1.7

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

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

Step 4.5.2

Multiply $2$ by $1$ .

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

Step 4.5.3

Change the $\pm$ to $-$ .

$x = \frac{- y - y i \sqrt{3}}{2}$

Step 4.5.4

Simplify the numerator.

Tap for more steps...

Step 4.5.4.1

Factor $y$ out of $- y - y i \sqrt{3}$ .

Tap for more steps...

Step 4.5.4.1.1

Factor $y$ out of $- y$ .

$x = \frac{y \cdot - 1 - y i \sqrt{3}}{2}$

Step 4.5.4.1.2

Factor $y$ out of $- y i \sqrt{3}$ .

$x = \frac{y \cdot - 1 + y (- 1 i \sqrt{3})}{2}$

Step 4.5.4.1.3

Factor $y$ out of $y \cdot - 1 + y (- 1 i \sqrt{3})$ .

$x = \frac{y (- 1 - 1 i \sqrt{3})}{2}$

Step 4.5.4.2

Rewrite $- 1 i$ as $- i$ .

$x = \frac{y (- 1 - i \sqrt{3})}{2}$

Step 4.5.5

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

$x = \frac{y (- 1 \cdot 1 - i \sqrt{3})}{2}$

Step 4.5.6

Factor $- 1$ out of $- i \sqrt{3}$ .

$x = \frac{y (- 1 \cdot 1 - (i \sqrt{3}))}{2}$

Step 4.5.7

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

$x = \frac{y (- 1 (1 + i \sqrt{3}))}{2}$

Step 4.5.8

Move the negative in front of the fraction.

$x = - \frac{y (1 + i \sqrt{3})}{2}$

Step 4.6

The final answer is the combination of both solutions.

$x = - \frac{y (1 - i \sqrt{3})}{2}$

$x = - \frac{y (1 + i \sqrt{3})}{2}$

$x = - \frac{y (1 - i \sqrt{3})}{2}$

$x = - \frac{y (1 + i \sqrt{3})}{2}$

Step 5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$