Solve 2x^2-0.015x+0.000025=0

$$2 x ^ { 2 } - 0.015 x + 0.000025 = 0$$

Solve for x

$$x=0.005$$

$$x=0.0025$$

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.

$$2x^{2}-0.015x+0.000025=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -0.015 for b, and 0.000025 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.015\right)±\sqrt{\left(-0.015\right)^{2}-4\times 2\times 0.000025}}{2\times 2}$$

Square -0.015 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.015\right)±\sqrt{0.000225-4\times 2\times 0.000025}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-0.015\right)±\sqrt{0.000225-8\times 0.000025}}{2\times 2}$$

Multiply -8 times 0.000025.

$$x=\frac{-\left(-0.015\right)±\sqrt{0.000225-0.0002}}{2\times 2}$$

Add 0.000225 to -0.0002 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.015\right)±\sqrt{0.000025}}{2\times 2}$$

Take the square root of 0.000025.

$$x=\frac{-\left(-0.015\right)±\frac{1}{200}}{2\times 2}$$

The opposite of -0.015 is 0.015.

$$x=\frac{0.015±\frac{1}{200}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{0.015±\frac{1}{200}}{4}$$

Now solve the equation x=\frac{0.015±\frac{1}{200}}{4} when ± is plus. Add 0.015 to \frac{1}{200} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{1}{50}}{4}$$

Divide \frac{1}{50} by 4.

$$x=\frac{1}{200}$$

Now solve the equation x=\frac{0.015±\frac{1}{200}}{4} when ± is minus. Subtract \frac{1}{200} from 0.015 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{1}{100}}{4}$$

Divide \frac{1}{100} by 4.

$$x=\frac{1}{400}$$

The equation is now solved.

$$x=\frac{1}{200} x=\frac{1}{400}$$

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.

$$2x^{2}-0.015x+0.000025=0$$

Subtract 0.000025 from both sides of the equation.

$$2x^{2}-0.015x+0.000025-0.000025=-0.000025$$

Subtracting 0.000025 from itself leaves 0.

$$2x^{2}-0.015x=-0.000025$$

Divide both sides by 2.

$$\frac{2x^{2}-0.015x}{2}=-\frac{0.000025}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{0.015}{2}\right)x=-\frac{0.000025}{2}$$

Divide -0.015 by 2.

$$x^{2}-0.0075x=-\frac{0.000025}{2}$$

Divide -0.000025 by 2.

$$x^{2}-0.0075x=-0.0000125$$

Divide -0.0075, the coefficient of the x term, by 2 to get -0.00375. Then add the square of -0.00375 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.0075x+\left(-0.00375\right)^{2}=-0.0000125+\left(-0.00375\right)^{2}$$

Square -0.00375 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.0075x+0.0000140625=-0.0000125+0.0000140625$$

Add -0.0000125 to 0.0000140625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.0075x+0.0000140625=0.0000015625$$

Factor x^{2}-0.0075x+0.0000140625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.00375\right)^{2}=0.0000015625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.00375\right)^{2}}=\sqrt{0.0000015625}$$

Simplify.

$$x-0.00375=\frac{1}{800} x-0.00375=-\frac{1}{800}$$

Add 0.00375 to both sides of the equation.

$$x=\frac{1}{200} x=\frac{1}{400}$$