Precalculus Examples

$\frac{3 x + 6 y}{x^{2} - y^{2}} \div \frac{5 x + 10 y}{x^{2} - 2 x y + y^{2}}$

Step 1

Set the denominator in $\frac{3 x + 6 y}{x^{2} - y^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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Step 2.1

Add $y^{2}$ to both sides of the equation.

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

Step 2.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x | = | y |$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm | y |$

Step 2.3.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.2.1

First, use the positive value of the $\pm$ to find the first solution.

$x = | y |$

Step 2.3.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - | y |$

Step 2.3.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = | y |$

$x = - | y |$

$x = | y |$

$x = - | y |$

$x = | y |$

$x = - | y |$

$x = | y |$

$x = - | y |$

Step 3

Set the denominator in $\frac{5 x + 10 y}{x^{2} - 2 x y + y^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 4

Solve for $x$ .

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

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

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Rewrite $4 \cdot 1 \cdot y^{2}$ as ${(2 \cdot 1 y)}^{2}$ .

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

Step 4.3.1.2

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

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

Step 4.3.1.3

Simplify.

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Step 4.3.1.3.1

Factor $2 y$ out of $- 2 y + 2 \cdot 1 y$ .

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Step 4.3.1.3.1.1

Factor $2 y$ out of $- 2 y$ .

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

Step 4.3.1.3.1.2

Factor $2 y$ out of $2 \cdot 1 y$ .

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

Step 4.3.1.3.1.3

Factor $2 y$ out of $2 y (- 1) + 2 y (1)$ .

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

Step 4.3.1.3.2

Add $- 1$ and $1$ .

$x = \frac{2 y \pm \sqrt{2 y \cdot (0 (- 2 y - (2 \cdot (1 y))))}}{2 \cdot 1}$

Step 4.3.1.3.3

Combine exponents.

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Step 4.3.1.3.3.1

Multiply $0$ by $2$ .

$x = \frac{2 y \pm \sqrt{0 y (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 4.3.1.3.3.2

Multiply $0$ by $y$ .

$x = \frac{2 y \pm \sqrt{0 (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 4.3.1.3.4

Factor $y$ out of $- 2 y - 1 \cdot 2 \cdot 1 y$ .

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Step 4.3.1.3.4.1

Factor $y$ out of $- 2 y$ .

$x = \frac{2 y \pm \sqrt{0 (y \cdot - 2 - 1 \cdot 2 \cdot (1 y))}}{2 \cdot 1}$

Step 4.3.1.3.4.2

Factor $y$ out of $- 1 \cdot 2 \cdot 1 y$ .

$x = \frac{2 y \pm \sqrt{0 (y \cdot - 2 + y (- 1 \cdot 2 \cdot 1))}}{2 \cdot 1}$

Step 4.3.1.3.4.3

Factor $y$ out of $y \cdot - 2 + y (- 1 \cdot 2 \cdot 1)$ .

$x = \frac{2 y \pm \sqrt{0 (y (- 2 - 1 \cdot 2 \cdot 1))}}{2 \cdot 1}$

Step 4.3.1.3.5

Multiply $- 1 \cdot 2 \cdot 1$ .

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Step 4.3.1.3.5.1

Multiply $- 1$ by $2$ .

$x = \frac{2 y \pm \sqrt{0 (y (- 2 - 2 \cdot 1))}}{2 \cdot 1}$

Step 4.3.1.3.5.2

Multiply $- 2$ by $1$ .

$x = \frac{2 y \pm \sqrt{0 (y (- 2 - 2))}}{2 \cdot 1}$

Step 4.3.1.3.6

Subtract $2$ from $- 2$ .

$x = \frac{2 y \pm \sqrt{0 y \cdot - 4}}{2 \cdot 1}$

Step 4.3.1.3.7

Combine exponents.

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Step 4.3.1.3.7.1

Multiply $0$ by $y$ .

$x = \frac{2 y \pm \sqrt{0 \cdot - 4}}{2 \cdot 1}$

Step 4.3.1.3.7.2

Multiply $0$ by $- 4$ .

$x = \frac{2 y \pm \sqrt{0}}{2 \cdot 1}$

Step 4.3.1.4

Rewrite $0$ as $0^{2}$ .

$x = \frac{2 y \pm \sqrt{0^{2}}}{2 \cdot 1}$

Step 4.3.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 y \pm 0}{2 \cdot 1}$

Step 4.3.1.6

$2 y$ plus or minus $0$ is $2 y$ .

$x = \frac{2 y}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{2 y}{2}$

Step 4.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.1

Cancel the common factor.

$x = \frac{2 y}{2}$

Step 4.3.3.2

Divide $y$ by $1$ .

$x = y$

Step 4.4

The final answer is the combination of both solutions.

$x = y$ Double roots

Step 5

Set the denominator in $\frac{3 x + 6 y}{x^{2} - y^{2}} \div \frac{5 x + 10 y}{x^{2} - 2 x y + y^{2}}$ equal to $0$ to find where the expression is undefined.

$\frac{5 x + 10 y}{x^{2} - 2 x y + y^{2}} = 0$

Step 6

Solve for $x$ .

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Step 6.1

Set the numerator equal to zero.

$5 x + 10 y = 0$

Step 6.2

Solve the equation for $x$ .

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Step 6.2.1

Subtract $10 y$ from both sides of the equation.

$5 x = - 10 y$

Step 6.2.2

Divide each term in $5 x = - 10 y$ by $5$ and simplify.

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Step 6.2.2.1

Divide each term in $5 x = - 10 y$ by $5$ .

$\frac{5 x}{5} = \frac{- 10 y}{5}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $5$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 10 y}{5}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10 y}{5}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Cancel the common factor of $- 10$ and $5$ .

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Step 6.2.2.3.1.1

Factor $5$ out of $- 10 y$ .

$x = \frac{5 (- 2 y)}{5}$

Step 6.2.2.3.1.2

Cancel the common factors.

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Step 6.2.2.3.1.2.1

Factor $5$ out of $5$ .

$x = \frac{5 (- 2 y)}{5 (1)}$

Step 6.2.2.3.1.2.2

Cancel the common factor.

$x = \frac{5 (- 2 y)}{5 \cdot 1}$

Step 6.2.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 2 y}{1}$

Step 6.2.2.3.1.2.4

Divide $- 2 y$ by $1$ .

$x = - 2 y$

Step 7

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$