Solve x^2+0.00083x-0.00415=0

$${ x }^{ 2 } +0.00083x-0.00415=0$$

Solve for x

$$x=\frac{\sqrt{166006889}-83}{200000}\approx 0.06400683$$

$$x=\frac{-\sqrt{166006889}-83}{200000}\approx -0.06483683$$

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.

$$x^{2}+0.00083x-0.00415=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0.00083 for b, and -0.00415 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.00083±\sqrt{0.00083^{2}-4\left(-0.00415\right)}}{2}$$

Square 0.00083 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.00083±\sqrt{0.0000006889-4\left(-0.00415\right)}}{2}$$

Multiply -4 times -0.00415.

$$x=\frac{-0.00083±\sqrt{0.0000006889+0.0166}}{2}$$

Add 0.0000006889 to 0.0166 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.00083±\sqrt{0.0166006889}}{2}$$

Take the square root of 0.0166006889.

$$x=\frac{-0.00083±\frac{\sqrt{166006889}}{100000}}{2}$$

Now solve the equation x=\frac{-0.00083±\frac{\sqrt{166006889}}{100000}}{2} when ± is plus. Add -0.00083 to \frac{\sqrt{166006889}}{100000}.

$$x=\frac{\sqrt{166006889}-83}{2\times 100000}$$

Divide \frac{-83+\sqrt{166006889}}{100000} by 2.

$$x=\frac{\sqrt{166006889}-83}{200000}$$

Now solve the equation x=\frac{-0.00083±\frac{\sqrt{166006889}}{100000}}{2} when ± is minus. Subtract \frac{\sqrt{166006889}}{100000} from -0.00083.

$$x=\frac{-\sqrt{166006889}-83}{2\times 100000}$$

Divide \frac{-83-\sqrt{166006889}}{100000} by 2.

$$x=\frac{-\sqrt{166006889}-83}{200000}$$

The equation is now solved.

$$x=\frac{\sqrt{166006889}-83}{200000} x=\frac{-\sqrt{166006889}-83}{200000}$$

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.

$$x^{2}+0.00083x-0.00415=0$$

Add 0.00415 to both sides of the equation.

$$x^{2}+0.00083x-0.00415-\left(-0.00415\right)=-\left(-0.00415\right)$$

Subtracting -0.00415 from itself leaves 0.

$$x^{2}+0.00083x=-\left(-0.00415\right)$$

Subtract -0.00415 from 0.

$$x^{2}+0.00083x=0.00415$$

Divide 0.00083, the coefficient of the x term, by 2 to get 0.000415. Then add the square of 0.000415 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.00083x+0.000415^{2}=0.00415+0.000415^{2}$$

Square 0.000415 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.00083x+0.000000172225=0.00415+0.000000172225$$

Add 0.00415 to 0.000000172225 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+0.00083x+0.000000172225=0.004150172225$$

Factor x^{2}+0.00083x+0.000000172225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.000415\right)^{2}=0.004150172225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.000415\right)^{2}}=\sqrt{0.004150172225}$$

Simplify.

$$x+0.000415=\frac{\sqrt{166006889}}{200000} x+0.000415=-\frac{\sqrt{166006889}}{200000}$$

Subtract 0.000415 from both sides of the equation.

$$x=\frac{\sqrt{166006889}-83}{200000} x=\frac{-\sqrt{166006889}-83}{200000}$$