Solve x^2+180x-0.10824=0

$${ x }^{ 2 } +180x-0.10824=0$$

Solve for x

$$x=\frac{\sqrt{506256765}}{250}-90\approx 0.000601331$$

$$x=-\frac{\sqrt{506256765}}{250}-90\approx -180.000601331$$

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.

$$x^{2}+180x-0.10824=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 180 for b, and -0.10824 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-180±\sqrt{180^{2}-4\left(-0.10824\right)}}{2}$$

Square 180.

$$x=\frac{-180±\sqrt{32400-4\left(-0.10824\right)}}{2}$$

Multiply -4 times -0.10824.

$$x=\frac{-180±\sqrt{32400+0.43296}}{2}$$

Add 32400 to 0.43296.

$$x=\frac{-180±\sqrt{32400.43296}}{2}$$

Take the square root of 32400.43296.

$$x=\frac{-180±\frac{\sqrt{506256765}}{125}}{2}$$

Now solve the equation x=\frac{-180±\frac{\sqrt{506256765}}{125}}{2} when ± is plus. Add -180 to \frac{\sqrt{506256765}}{125}.

$$x=\frac{\frac{\sqrt{506256765}}{125}-180}{2}$$

Divide -180+\frac{\sqrt{506256765}}{125} by 2.

$$x=\frac{\sqrt{506256765}}{250}-90$$

Now solve the equation x=\frac{-180±\frac{\sqrt{506256765}}{125}}{2} when ± is minus. Subtract \frac{\sqrt{506256765}}{125} from -180.

$$x=\frac{-\frac{\sqrt{506256765}}{125}-180}{2}$$

Divide -180-\frac{\sqrt{506256765}}{125} by 2.

$$x=-\frac{\sqrt{506256765}}{250}-90$$

The equation is now solved.

$$x=\frac{\sqrt{506256765}}{250}-90 x=-\frac{\sqrt{506256765}}{250}-90$$

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.

$$x^{2}+180x-0.10824=0$$

Add 0.10824 to both sides of the equation.

$$x^{2}+180x-0.10824-\left(-0.10824\right)=-\left(-0.10824\right)$$

Subtracting -0.10824 from itself leaves 0.

$$x^{2}+180x=-\left(-0.10824\right)$$

Subtract -0.10824 from 0.

$$x^{2}+180x=0.10824$$

Divide 180, the coefficient of the x term, by 2 to get 90. Then add the square of 90 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+180x+90^{2}=0.10824+90^{2}$$

Square 90.

$$x^{2}+180x+8100=0.10824+8100$$

Add 0.10824 to 8100.

$$x^{2}+180x+8100=8100.10824$$

Factor x^{2}+180x+8100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+90\right)^{2}=8100.10824$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+90\right)^{2}}=\sqrt{8100.10824}$$

Simplify.

$$x+90=\frac{\sqrt{506256765}}{250} x+90=-\frac{\sqrt{506256765}}{250}$$

Subtract 90 from both sides of the equation.

$$x=\frac{\sqrt{506256765}}{250}-90 x=-\frac{\sqrt{506256765}}{250}-90$$