Solve x^2+84x-160=0

$${ x }^{ 2 } +84x-160=0$$

Solve for x

$$x=2\sqrt{481}-42\approx 1.863424399$$

$$x=-2\sqrt{481}-42\approx -85.863424399$$

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.

$$x^{2}+84x-160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 84 for b, and -160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-84±\sqrt{84^{2}-4\left(-160\right)}}{2}$$

Square 84.

$$x=\frac{-84±\sqrt{7056-4\left(-160\right)}}{2}$$

Multiply -4 times -160.

$$x=\frac{-84±\sqrt{7056+640}}{2}$$

Add 7056 to 640.

$$x=\frac{-84±\sqrt{7696}}{2}$$

Take the square root of 7696.

$$x=\frac{-84±4\sqrt{481}}{2}$$

Now solve the equation x=\frac{-84±4\sqrt{481}}{2} when ± is plus. Add -84 to 4\sqrt{481}.

$$x=\frac{4\sqrt{481}-84}{2}$$

Divide -84+4\sqrt{481} by 2.

$$x=2\sqrt{481}-42$$

Now solve the equation x=\frac{-84±4\sqrt{481}}{2} when ± is minus. Subtract 4\sqrt{481} from -84.

$$x=\frac{-4\sqrt{481}-84}{2}$$

Divide -84-4\sqrt{481} by 2.

$$x=-2\sqrt{481}-42$$

The equation is now solved.

$$x=2\sqrt{481}-42 x=-2\sqrt{481}-42$$

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.

$$x^{2}+84x-160=0$$

Add 160 to both sides of the equation.

$$x^{2}+84x-160-\left(-160\right)=-\left(-160\right)$$

Subtracting -160 from itself leaves 0.

$$x^{2}+84x=-\left(-160\right)$$

Subtract -160 from 0.

$$x^{2}+84x=160$$

Divide 84, the coefficient of the x term, by 2 to get 42. Then add the square of 42 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+84x+42^{2}=160+42^{2}$$

Square 42.

$$x^{2}+84x+1764=160+1764$$

Add 160 to 1764.

$$x^{2}+84x+1764=1924$$

Factor x^{2}+84x+1764. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+42\right)^{2}=1924$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+42\right)^{2}}=\sqrt{1924}$$

Simplify.

$$x+42=2\sqrt{481} x+42=-2\sqrt{481}$$

Subtract 42 from both sides of the equation.

$$x=2\sqrt{481}-42 x=-2\sqrt{481}-42$$