Algebra Examples

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

Step 1

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 1.1

Simplify the left side.

Tap for more steps...

Step 1.1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

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

Subtract $11$ from both sides of the equation.

$\frac{(x + 1) (x - 1)}{2} - 11 x - 11 = 0$

Step 2

Multiply through by the least common denominator $2$ , then simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

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

Tap for more steps...

Step 2.4.1.1

Apply the distributive property.

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

Step 2.4.1.2

Apply the distributive property.

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

Step 2.4.1.3

Apply the distributive property.

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

Step 2.4.2

Simplify and combine like terms.

Tap for more steps...

Step 2.4.2.1

Simplify each term.

Tap for more steps...

Step 2.4.2.1.1

Multiply $x$ by $x$ .

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

Step 2.4.2.1.2

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

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

Step 2.4.2.1.3

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

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

Step 2.4.2.1.4

Multiply $x$ by $1$ .

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

Step 2.4.2.1.5

Multiply $- 1$ by $1$ .

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

Step 2.4.2.2

Add $- x$ and $x$ .

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

Step 2.4.2.3

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

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

Step 2.4.3

Multiply $2$ by $- 11$ .

$2 (\frac{x^{2} - 1 - 22 x}{2} - 11) = 0$

Step 2.4.4

Reorder terms.

$2 (\frac{x^{2} - 22 x - 1}{2} - 11) = 0$

Step 2.5

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

$2 (\frac{x^{2} - 22 x - 1}{2} - 11 \cdot \frac{2}{2}) = 0$

Step 2.6

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

$2 (\frac{x^{2} - 22 x - 1}{2} + \frac{- 11 \cdot 2}{2}) = 0$

Step 2.7

Combine the numerators over the common denominator.

$2 (\frac{x^{2} - 22 x - 1 - 11 \cdot 2}{2}) = 0$

Step 2.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.8.1

Cancel the common factor.

$2 (\frac{x^{2} - 22 x - 1 - 11 \cdot 2}{2}) = 0$

Step 2.8.2

Rewrite the expression.

$x^{2} - 22 x - 1 - 11 \cdot 2 = 0$

Step 2.9

Multiply $- 11$ by $2$ .

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

Step 2.10

Subtract $22$ from $- 1$ .

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 22$ , and $c = - 23$ into the quadratic formula and solve for $x$ .

$\frac{22 \pm \sqrt{{(- 22)}^{2} - 4 \cdot (1 \cdot - 23)}}{2 \cdot 1}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Raise $- 22$ to the power of $2$ .

$x = \frac{22 \pm \sqrt{484 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 23$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{22 \pm \sqrt{484 - 4 \cdot - 23}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 23$ .

$x = \frac{22 \pm \sqrt{484 + 92}}{2 \cdot 1}$

Step 5.1.3

Add $484$ and $92$ .

$x = \frac{22 \pm \sqrt{576}}{2 \cdot 1}$

Step 5.1.4

Rewrite $576$ as $24^{2}$ .

$x = \frac{22 \pm \sqrt{24^{2}}}{2 \cdot 1}$

Step 5.1.5

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

$x = \frac{22 \pm 24}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{22 \pm 24}{2}$

Step 5.3

Simplify $\frac{22 \pm 24}{2}$ .

$x = 11 \pm 12$

Step 6

The final answer is the combination of both solutions.

$x = 23, - 1$