Solve Using the Quadratic Formula (x^2-1)/2-11x=11

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

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

Simplify the left side.

Simplify the numerator.

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

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

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 $b = 1$ .

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

Subtract $11$ from both sides of the equation.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Add $- x$ and $x$ .

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

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

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

Multiply $2$ by $- 11$ .

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

Reorder terms.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 11$ by $2$ .

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

Subtract $22$ from $- 1$ .

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 23$ .

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

Add $484$ and $92$ .

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

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

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

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

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

Multiply $2$ by $1$ .

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

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

$x = 11 \pm 12$

The final answer is the combination of both solutions.

$x = 23, - 1$

Solve Using the Quadratic Formula (x^2-1)/2-11x=11

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