Solve 11=x^2-4x

$$11 = x ^ { 2 } - 4 x$$

Solve for x

$$x=\sqrt{15}+2\approx 5.872983346$$

$$x=2-\sqrt{15}\approx -1.872983346$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-4x=11$$

Subtract 11 from both sides.

$$x^{2}-4x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-11\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-11\right)}}{2}$$

Multiply -4 times -11.

$$x=\frac{-\left(-4\right)±\sqrt{16+44}}{2}$$

Add 16 to 44.

$$x=\frac{-\left(-4\right)±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{-\left(-4\right)±2\sqrt{15}}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{4±2\sqrt{15}}{2} when ± is plus. Add 4 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}+4}{2}$$

Divide 4+2\sqrt{15} by 2.

$$x=\sqrt{15}+2$$

Now solve the equation x=\frac{4±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from 4.

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

Divide 4-2\sqrt{15} by 2.

$$x=2-\sqrt{15}$$

The equation is now solved.

$$x=\sqrt{15}+2 x=2-\sqrt{15}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-4x=11$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=11+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=11+4$$

Add 11 to 4.

$$x^{2}-4x+4=15$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{15}$$

Simplify.

$$x-2=\sqrt{15} x-2=-\sqrt{15}$$

Add 2 to both sides of the equation.

$$x=\sqrt{15}+2 x=2-\sqrt{15}$$