Solve 3x+209xx=0.001

$$3 x + 209 x x = 0.001$$

Solve for x

$$x=\frac{\sqrt{24590}}{20900}-\frac{3}{418}\approx 0.000325933$$

$$x=-\frac{\sqrt{24590}}{20900}-\frac{3}{418}\approx -0.014679999$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$3x+209x^{2}=0.001$$

Subtract 0.001 from both sides.

$$3x+209x^{2}-0.001=0$$

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.

$$209x^{2}+3x-0.001=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 209 for a, 3 for b, and -0.001 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 209\left(-0.001\right)}}{2\times 209}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 209\left(-0.001\right)}}{2\times 209}$$

Multiply -4 times 209.

$$x=\frac{-3±\sqrt{9-836\left(-0.001\right)}}{2\times 209}$$

Multiply -836 times -0.001.

$$x=\frac{-3±\sqrt{9+0.836}}{2\times 209}$$

Add 9 to 0.836.

$$x=\frac{-3±\sqrt{9.836}}{2\times 209}$$

Take the square root of 9.836.

$$x=\frac{-3±\frac{\sqrt{24590}}{50}}{2\times 209}$$

Multiply 2 times 209.

$$x=\frac{-3±\frac{\sqrt{24590}}{50}}{418}$$

Now solve the equation x=\frac{-3±\frac{\sqrt{24590}}{50}}{418} when ± is plus. Add -3 to \frac{\sqrt{24590}}{50}.

$$x=\frac{\frac{\sqrt{24590}}{50}-3}{418}$$

Divide -3+\frac{\sqrt{24590}}{50} by 418.

$$x=\frac{\sqrt{24590}}{20900}-\frac{3}{418}$$

Now solve the equation x=\frac{-3±\frac{\sqrt{24590}}{50}}{418} when ± is minus. Subtract \frac{\sqrt{24590}}{50} from -3.

$$x=\frac{-\frac{\sqrt{24590}}{50}-3}{418}$$

Divide -3-\frac{\sqrt{24590}}{50} by 418.

$$x=-\frac{\sqrt{24590}}{20900}-\frac{3}{418}$$

The equation is now solved.

$$x=\frac{\sqrt{24590}}{20900}-\frac{3}{418} x=-\frac{\sqrt{24590}}{20900}-\frac{3}{418}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$3x+209x^{2}=0.001$$

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.

$$209x^{2}+3x=0.001$$

Divide both sides by 209.

$$\frac{209x^{2}+3x}{209}=\frac{0.001}{209}$$

Dividing by 209 undoes the multiplication by 209.

$$x^{2}+\frac{3}{209}x=\frac{0.001}{209}$$

Divide 0.001 by 209.

$$x^{2}+\frac{3}{209}x=\frac{1}{209000}$$

Divide \frac{3}{209}, the coefficient of the x term, by 2 to get \frac{3}{418}. Then add the square of \frac{3}{418} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{209}x+\left(\frac{3}{418}\right)^{2}=\frac{1}{209000}+\left(\frac{3}{418}\right)^{2}$$

Square \frac{3}{418} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{209}x+\frac{9}{174724}=\frac{1}{209000}+\frac{9}{174724}$$

Add \frac{1}{209000} to \frac{9}{174724} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{209}x+\frac{9}{174724}=\frac{2459}{43681000}$$

Factor x^{2}+\frac{3}{209}x+\frac{9}{174724}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{418}\right)^{2}=\frac{2459}{43681000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{418}\right)^{2}}=\sqrt{\frac{2459}{43681000}}$$

Simplify.

$$x+\frac{3}{418}=\frac{\sqrt{24590}}{20900} x+\frac{3}{418}=-\frac{\sqrt{24590}}{20900}$$

Subtract \frac{3}{418} from both sides of the equation.

$$x=\frac{\sqrt{24590}}{20900}-\frac{3}{418} x=-\frac{\sqrt{24590}}{20900}-\frac{3}{418}$$