Solve x^2-4x-5=11

$${ x }^{ 2 } -4x-5 = 11$$

Solve for x

$$x=2\sqrt{5}+2\approx 6.472135955$$

$$x=2-2\sqrt{5}\approx -2.472135955$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-4x-5=11$$

Subtract 11 from both sides of the equation.

$$x^{2}-4x-5-11=11-11$$

Subtracting 11 from itself leaves 0.

$$x^{2}-4x-5-11=0$$

Subtract 11 from -5.

$$x^{2}-4x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

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

Multiply -4 times -16.

$$x=\frac{-\left(-4\right)±\sqrt{16+64}}{2}$$

Add 16 to 64.

$$x=\frac{-\left(-4\right)±\sqrt{80}}{2}$$

Take the square root of 80.

$$x=\frac{-\left(-4\right)±4\sqrt{5}}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{5}}{2}$$

Now solve the equation x=\frac{4±4\sqrt{5}}{2} when ± is plus. Add 4 to 4\sqrt{5}.

$$x=\frac{4\sqrt{5}+4}{2}$$

Divide 4+4\sqrt{5} by 2.

$$x=2\sqrt{5}+2$$

Now solve the equation x=\frac{4±4\sqrt{5}}{2} when ± is minus. Subtract 4\sqrt{5} from 4.

$$x=\frac{4-4\sqrt{5}}{2}$$

Divide 4-4\sqrt{5} by 2.

$$x=2-2\sqrt{5}$$

The equation is now solved.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-4x-5=11$$

Add 5 to both sides of the equation.

$$x^{2}-4x-5-\left(-5\right)=11-\left(-5\right)$$

Subtracting -5 from itself leaves 0.

$$x^{2}-4x=11-\left(-5\right)$$

Subtract -5 from 11.

$$x^{2}-4x=16$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=16+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=16+4$$

Add 16 to 4.

$$x^{2}-4x+4=20$$

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}.

$$\left(x-2\right)^{2}=20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{20}$$

Simplify.

$$x-2=2\sqrt{5} x-2=-2\sqrt{5}$$

Add 2 to both sides of the equation.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$