Solve Using the Quadratic Formula x(x-1)=12

$x (x - 1) = 12$

Step 1

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

Simplify the left side.

Simplify $x (x - 1)$ .

Simplify by multiplying through.

Apply the distributive property.

$x \cdot x + x \cdot - 1 = 12$

Simplify the expression.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 = 12$

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

$x^{2} - 1 \cdot x = 12$

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

$x^{2} - x = 12$

Subtract $12$ from both sides of the equation.

$x^{2} - x - 12 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{1 \pm \sqrt{1 + 48}}{2 \cdot 1}$

Add $1$ and $48$ .

$x = \frac{1 \pm \sqrt{49}}{2 \cdot 1}$

Rewrite $49$ as $7^{2}$ .

$x = \frac{1 \pm \sqrt{7^{2}}}{2 \cdot 1}$

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

$x = \frac{1 \pm 7}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm 7}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 4, - 3$