Algebra Examples

${(x + 1)}^{2} - 2 = \frac{2}{x}$

Step 1

Add $2$ to both sides of the equation.

${(x + 1)}^{2} = \frac{2}{x} + 2$

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.

$1, x, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in ${(x + 1)}^{2} = \frac{2}{x} + 2$ by $x$ to eliminate the fractions.

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Step 3.1

Multiply each term in ${(x + 1)}^{2} = \frac{2}{x} + 2$ by $x$ .

${(x + 1)}^{2} x = \frac{2}{x} x + 2 x$

Step 3.2

Simplify the left side.

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Step 3.2.1

Reorder factors in ${(x + 1)}^{2} x$ .

$x {(x + 1)}^{2} = \frac{2}{x} x + 2 x$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

Cancel the common factor.

$x {(x + 1)}^{2} = \frac{2}{x} x + 2 x$

Step 3.3.1.2

Rewrite the expression.

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

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $2 x$ from both sides of the equation.

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

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$x ((x + 1) (x + 1)) - 2 x = 2$

Step 4.1.2.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 4.1.2.2.1

Apply the distributive property.

$x (x (x + 1) + 1 (x + 1)) - 2 x = 2$

Step 4.1.2.2.2

Apply the distributive property.

$x (x \cdot x + x \cdot 1 + 1 (x + 1)) - 2 x = 2$

Step 4.1.2.2.3

Apply the distributive property.

$x (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) - 2 x = 2$

Step 4.1.2.3

Simplify and combine like terms.

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Step 4.1.2.3.1

Simplify each term.

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Step 4.1.2.3.1.1

Multiply $x$ by $x$ .

$x (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) - 2 x = 2$

Step 4.1.2.3.1.2

Multiply $x$ by $1$ .

$x (x^{2} + x + 1 x + 1 \cdot 1) - 2 x = 2$

Step 4.1.2.3.1.3

Multiply $x$ by $1$ .

$x (x^{2} + x + x + 1 \cdot 1) - 2 x = 2$

Step 4.1.2.3.1.4

Multiply $1$ by $1$ .

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

Step 4.1.2.3.2

Add $x$ and $x$ .

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

Step 4.1.2.4

Apply the distributive property.

$x \cdot x^{2} + x (2 x) + x \cdot 1 - 2 x = 2$

Step 4.1.2.5

Simplify.

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Step 4.1.2.5.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.1.2.5.1.1

Multiply $x$ by $x^{2}$ .

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Step 4.1.2.5.1.1.1

Raise $x$ to the power of $1$ .

$x^{1} x^{2} + x (2 x) + x \cdot 1 - 2 x = 2$

Step 4.1.2.5.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 2} + x (2 x) + x \cdot 1 - 2 x = 2$

Step 4.1.2.5.1.2

Add $1$ and $2$ .

$x^{3} + x (2 x) + x \cdot 1 - 2 x = 2$

Step 4.1.2.5.2

Rewrite using the commutative property of multiplication.

$x^{3} + 2 x \cdot x + x \cdot 1 - 2 x = 2$

Step 4.1.2.5.3

Multiply $x$ by $1$ .

$x^{3} + 2 x \cdot x + x - 2 x = 2$

Step 4.1.2.6

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.2.6.1

Move $x$ .

$x^{3} + 2 (x \cdot x) + x - 2 x = 2$

Step 4.1.2.6.2

Multiply $x$ by $x$ .

$x^{3} + 2 x^{2} + x - 2 x = 2$

Step 4.1.3

Subtract $2 x$ from $x$ .

$x^{3} + 2 x^{2} - x = 2$

Step 4.2

Subtract $2$ from both sides of the equation.

$x^{3} + 2 x^{2} - x - 2 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor out the greatest common factor from each group.

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Step 4.3.1.1

Group the first two terms and the last two terms.

$(x^{3} + 2 x^{2}) - x - 2 = 0$

Step 4.3.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.3.2

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

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

Step 4.3.3

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

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

Step 4.3.4

Factor.

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Step 4.3.4.1

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 $b = 1$ .

$(x + 2) ((x + 1) (x - 1)) = 0$

Step 4.3.4.2

Remove unnecessary parentheses.

$(x + 2) (x + 1) (x - 1) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 4.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.6.1

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

$x + 1 = 0$

Step 4.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.7

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.7.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.8

The final solution is all the values that make $(x + 2) (x + 1) (x - 1) = 0$ true.

$x = - 2, - 1, 1$