Solve x^2+985565x+2=0

$${ x }^{ 2 } +985565x+2=0$$

Solve for x

$$x=\frac{\sqrt{971338369217}-985565}{2}\approx -0.000002029$$

$$x=\frac{-\sqrt{971338369217}-985565}{2}\approx -985564.999997971$$

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.

$$x^{2}+985565x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 985565 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-985565±\sqrt{985565^{2}-4\times 2}}{2}$$

Square 985565.

$$x=\frac{-985565±\sqrt{971338369225-4\times 2}}{2}$$

Multiply -4 times 2.

$$x=\frac{-985565±\sqrt{971338369225-8}}{2}$$

Add 971338369225 to -8.

$$x=\frac{-985565±\sqrt{971338369217}}{2}$$

Now solve the equation x=\frac{-985565±\sqrt{971338369217}}{2} when ± is plus. Add -985565 to \sqrt{971338369217}.

$$x=\frac{\sqrt{971338369217}-985565}{2}$$

Now solve the equation x=\frac{-985565±\sqrt{971338369217}}{2} when ± is minus. Subtract \sqrt{971338369217} from -985565.

$$x=\frac{-\sqrt{971338369217}-985565}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{971338369217}-985565}{2} x=\frac{-\sqrt{971338369217}-985565}{2}$$

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.

$$x^{2}+985565x+2=0$$

Subtract 2 from both sides of the equation.

$$x^{2}+985565x+2-2=-2$$

Subtracting 2 from itself leaves 0.

$$x^{2}+985565x=-2$$

Divide 985565, the coefficient of the x term, by 2 to get \frac{985565}{2}. Then add the square of \frac{985565}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+985565x+\left(\frac{985565}{2}\right)^{2}=-2+\left(\frac{985565}{2}\right)^{2}$$

Square \frac{985565}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+985565x+\frac{971338369225}{4}=-2+\frac{971338369225}{4}$$

Add -2 to \frac{971338369225}{4}.

$$x^{2}+985565x+\frac{971338369225}{4}=\frac{971338369217}{4}$$

Factor x^{2}+985565x+\frac{971338369225}{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}.

$$\left(x+\frac{985565}{2}\right)^{2}=\frac{971338369217}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{985565}{2}\right)^{2}}=\sqrt{\frac{971338369217}{4}}$$

Simplify.

$$x+\frac{985565}{2}=\frac{\sqrt{971338369217}}{2} x+\frac{985565}{2}=-\frac{\sqrt{971338369217}}{2}$$

Subtract \frac{985565}{2} from both sides of the equation.

$$x=\frac{\sqrt{971338369217}-985565}{2} x=\frac{-\sqrt{971338369217}-985565}{2}$$