Solve x^2+120x-4000=0

$${ x }^{ 2 } +120x-4000 = 0$$

Solve for x

$$x=20\sqrt{19}-60\approx 27.177978871$$

$$x=-20\sqrt{19}-60\approx -147.177978871$$

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.

$$x^{2}+120x-4000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 120 for b, and -4000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-4000\right)}}{2}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\left(-4000\right)}}{2}$$

Multiply -4 times -4000.

$$x=\frac{-120±\sqrt{14400+16000}}{2}$$

Add 14400 to 16000.

$$x=\frac{-120±\sqrt{30400}}{2}$$

Take the square root of 30400.

$$x=\frac{-120±40\sqrt{19}}{2}$$

Now solve the equation x=\frac{-120±40\sqrt{19}}{2} when ± is plus. Add -120 to 40\sqrt{19}.

$$x=\frac{40\sqrt{19}-120}{2}$$

Divide -120+40\sqrt{19} by 2.

$$x=20\sqrt{19}-60$$

Now solve the equation x=\frac{-120±40\sqrt{19}}{2} when ± is minus. Subtract 40\sqrt{19} from -120.

$$x=\frac{-40\sqrt{19}-120}{2}$$

Divide -120-40\sqrt{19} by 2.

$$x=-20\sqrt{19}-60$$

The equation is now solved.

$$x=20\sqrt{19}-60 x=-20\sqrt{19}-60$$

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.

$$x^{2}+120x-4000=0$$

Add 4000 to both sides of the equation.

$$x^{2}+120x-4000-\left(-4000\right)=-\left(-4000\right)$$

Subtracting -4000 from itself leaves 0.

$$x^{2}+120x=-\left(-4000\right)$$

Subtract -4000 from 0.

$$x^{2}+120x=4000$$

Divide 120, the coefficient of the x term, by 2 to get 60. Then add the square of 60 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+120x+60^{2}=4000+60^{2}$$

Square 60.

$$x^{2}+120x+3600=4000+3600$$

Add 4000 to 3600.

$$x^{2}+120x+3600=7600$$

Factor x^{2}+120x+3600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+60\right)^{2}=7600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+60\right)^{2}}=\sqrt{7600}$$

Simplify.

$$x+60=20\sqrt{19} x+60=-20\sqrt{19}$$

Subtract 60 from both sides of the equation.

$$x=20\sqrt{19}-60 x=-20\sqrt{19}-60$$