Algebra Examples

$\frac{x^{2} - 1}{2} - 11 x = 11$

Step 1

Simplify $\frac{x^{2} - 1}{2} - 11 x$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

$\frac{x^{2} - 1^{2}}{2} - 11 x = 11$

Step 1.1.2

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

$\frac{(x + 1) (x - 1)}{2} - 11 x = 11$

Step 1.2

To write $- 11 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(x + 1) (x - 1)}{2} - 11 x \cdot \frac{2}{2} = 11$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $- 11 x$ and $\frac{2}{2}$ .

$\frac{(x + 1) (x - 1)}{2} + \frac{- 11 x \cdot 2}{2} = 11$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{(x + 1) (x - 1) - 11 x \cdot 2}{2} = 11$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

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Step 1.4.1.1

Apply the distributive property.

$\frac{x (x - 1) + 1 (x - 1) - 11 x \cdot 2}{2} = 11$

Step 1.4.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 + 1 (x - 1) - 11 x \cdot 2}{2} = 11$

Step 1.4.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.2

Simplify and combine like terms.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.2.1.2

Move $- 1$ to the left of $x$ .

$\frac{x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.2.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} - x + 1 x + 1 \cdot - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.2.1.4

Multiply $x$ by $1$ .

$\frac{x^{2} - x + x + 1 \cdot - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.2.1.5

Multiply $- 1$ by $1$ .

$\frac{x^{2} - x + x - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.2.2

Add $- x$ and $x$ .

$\frac{x^{2} + 0 - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.2.3

Add $x^{2}$ and $0$ .

$\frac{x^{2} - 1 - 11 x \cdot 2}{2} = 11$

Step 1.4.3

Multiply $2$ by $- 11$ .

$\frac{x^{2} - 1 - 22 x}{2} = 11$

Step 1.4.4

Reorder terms.

$\frac{x^{2} - 22 x - 1}{2} = 11$

Step 2

Multiply both sides by $2$ .

$\frac{x^{2} - 22 x - 1}{2} \cdot 2 = 11 \cdot 2$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{x^{2} - 22 x - 1}{2} \cdot 2 = 11 \cdot 2$

Step 3.1.1.2

Rewrite the expression.

$x^{2} - 22 x - 1 = 11 \cdot 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $11$ by $2$ .

$x^{2} - 22 x - 1 = 22$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $22$ from both sides of the equation.

$x^{2} - 22 x - 1 - 22 = 0$

Step 4.2

Subtract $22$ from $- 1$ .

$x^{2} - 22 x - 23 = 0$

Step 4.3

Factor $x^{2} - 22 x - 23$ using the AC method.

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Step 4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 23$ and whose sum is $- 22$ .

$- 23, 1$

Step 4.3.2

Write the factored form using these integers.

$(x - 23) (x + 1) = 0$

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 $0$ .

$x - 23 = 0$

$x + 1 = 0$

Step 4.5

Set $x - 23$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 23$ equal to $0$ .

$x - 23 = 0$

Step 4.5.2

Add $23$ to both sides of the equation.

$x = 23$

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

The final solution is all the values that make $(x - 23) (x + 1) = 0$ true.

$x = 23, - 1$