Solve Using the Quadratic Formula x(x-2)=15

$x (x - 2) = 15$

Step 1

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

Simplify the left side.

Simplify $x (x - 2)$ .

Apply the distributive property.

$x \cdot x + x \cdot - 2 = 15$

Simplify the expression.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 = 15$

Move $- 2$ to the left of $x$ .

$x^{2} - 2 x = 15$

Subtract $15$ from both sides of the equation.

$x^{2} - 2 x - 15 = 0$

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 $a = 1$ , $b = - 2$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 15)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 15$ .

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 15}}{2 \cdot 1}$

Multiply $- 4$ by $- 15$ .

$x = \frac{2 \pm \sqrt{4 + 60}}{2 \cdot 1}$

Add $4$ and $60$ .

$x = \frac{2 \pm \sqrt{64}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{2 \pm \sqrt{8^{2}}}{2 \cdot 1}$

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

$x = \frac{2 \pm 8}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{2 \pm 8}{2}$

Simplify $\frac{2 \pm 8}{2}$ .

$x = 1 \pm 4$

Step 5

The final answer is the combination of both solutions.

$x = 5, - 3$