Algebra Examples

$\frac{5 x^{2} + 10 x}{x^{2} + 2 x + 1} \div \frac{20 x + 40}{x^{2} - 1}$

Step 1

Set the denominator in $\frac{5 x^{2} + 10 x}{x^{2} + 2 x + 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 2 x + 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor using the perfect square rule.

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Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$x^{2} + 2 x + 1^{2} = 0$

Step 2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 2.1.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 1 + 1^{2} = 0$

Step 2.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

${(x + 1)}^{2} = 0$

Step 2.2

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3

Set the denominator in $\frac{20 x + 40}{x^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 1 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 4.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 4.4

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

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Step 4.4.1

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

$x = 1$

Step 4.4.2

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

$x = - 1$

Step 4.4.3

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

$x = 1, - 1$

Step 5

Set the denominator in $\frac{5 x^{2} + 10 x}{x^{2} + 2 x + 1} \div \frac{20 x + 40}{x^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$\frac{20 x + 40}{x^{2} - 1} = 0$

Step 6

Solve for $x$ .

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Step 6.1

Set the numerator equal to zero.

$20 x + 40 = 0$

Step 6.2

Solve the equation for $x$ .

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Step 6.2.1

Subtract $40$ from both sides of the equation.

$20 x = - 40$

Step 6.2.2

Divide each term in $20 x = - 40$ by $20$ and simplify.

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Step 6.2.2.1

Divide each term in $20 x = - 40$ by $20$ .

$\frac{20 x}{20} = \frac{- 40}{20}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $20$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{20 x}{20} = \frac{- 40}{20}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 40}{20}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Divide $- 40$ by $20$ .

$x = - 2$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = - 1, x = 1$

Step 8