Solve 0.03=100x-41666.662x^2

$$0.03 = 100 x - 41666.662 x ^ { 2 }$$

Solve for x

$$x=\frac{17\sqrt{1081315}+25000}{20833331}\approx 0.002048528$$

$$x=\frac{25000-17\sqrt{1081315}}{20833331}\approx 0.000351472$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$100x-41666.662x^{2}=0.03$$

Subtract 0.03 from both sides.

$$100x-41666.662x^{2}-0.03=0$$

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.

$$-41666.662x^{2}+100x-0.03=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -41666.662 for a, 100 for b, and -0.03 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-41666.662\right)\left(-0.03\right)}}{2\left(-41666.662\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-41666.662\right)\left(-0.03\right)}}{2\left(-41666.662\right)}$$

Multiply -4 times -41666.662.

$$x=\frac{-100±\sqrt{10000+166666.648\left(-0.03\right)}}{2\left(-41666.662\right)}$$

Multiply 166666.648 times -0.03 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-100±\sqrt{10000-4999.99944}}{2\left(-41666.662\right)}$$

Add 10000 to -4999.99944.

$$x=\frac{-100±\sqrt{5000.00056}}{2\left(-41666.662\right)}$$

Take the square root of 5000.00056.

$$x=\frac{-100±\frac{17\sqrt{1081315}}{250}}{2\left(-41666.662\right)}$$

Multiply 2 times -41666.662.

$$x=\frac{-100±\frac{17\sqrt{1081315}}{250}}{-83333.324}$$

Now solve the equation x=\frac{-100±\frac{17\sqrt{1081315}}{250}}{-83333.324} when ± is plus. Add -100 to \frac{17\sqrt{1081315}}{250}.

$$x=\frac{\frac{17\sqrt{1081315}}{250}-100}{-83333.324}$$

Divide -100+\frac{17\sqrt{1081315}}{250} by -83333.324 by multiplying -100+\frac{17\sqrt{1081315}}{250} by the reciprocal of -83333.324.

$$x=\frac{25000-17\sqrt{1081315}}{20833331}$$

Now solve the equation x=\frac{-100±\frac{17\sqrt{1081315}}{250}}{-83333.324} when ± is minus. Subtract \frac{17\sqrt{1081315}}{250} from -100.

$$x=\frac{-\frac{17\sqrt{1081315}}{250}-100}{-83333.324}$$

Divide -100-\frac{17\sqrt{1081315}}{250} by -83333.324 by multiplying -100-\frac{17\sqrt{1081315}}{250} by the reciprocal of -83333.324.

$$x=\frac{17\sqrt{1081315}+25000}{20833331}$$

The equation is now solved.

$$x=\frac{25000-17\sqrt{1081315}}{20833331} x=\frac{17\sqrt{1081315}+25000}{20833331}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$100x-41666.662x^{2}=0.03$$

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.

$$-41666.662x^{2}+100x=0.03$$

Divide both sides of the equation by -41666.662, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-41666.662x^{2}+100x}{-41666.662}=\frac{0.03}{-41666.662}$$

Dividing by -41666.662 undoes the multiplication by -41666.662.

$$x^{2}+\frac{100}{-41666.662}x=\frac{0.03}{-41666.662}$$

Divide 100 by -41666.662 by multiplying 100 by the reciprocal of -41666.662.

$$x^{2}-\frac{50000}{20833331}x=\frac{0.03}{-41666.662}$$

Divide 0.03 by -41666.662 by multiplying 0.03 by the reciprocal of -41666.662.

$$x^{2}-\frac{50000}{20833331}x=-\frac{15}{20833331}$$

Divide -\frac{50000}{20833331}, the coefficient of the x term, by 2 to get -\frac{25000}{20833331}. Then add the square of -\frac{25000}{20833331} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{50000}{20833331}x+\left(-\frac{25000}{20833331}\right)^{2}=-\frac{15}{20833331}+\left(-\frac{25000}{20833331}\right)^{2}$$

Square -\frac{25000}{20833331} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{50000}{20833331}x+\frac{625000000}{434027680555561}=-\frac{15}{20833331}+\frac{625000000}{434027680555561}$$

Add -\frac{15}{20833331} to \frac{625000000}{434027680555561} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{50000}{20833331}x+\frac{625000000}{434027680555561}=\frac{312500035}{434027680555561}$$

Factor x^{2}-\frac{50000}{20833331}x+\frac{625000000}{434027680555561}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25000}{20833331}\right)^{2}=\frac{312500035}{434027680555561}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25000}{20833331}\right)^{2}}=\sqrt{\frac{312500035}{434027680555561}}$$

Simplify.

$$x-\frac{25000}{20833331}=\frac{17\sqrt{1081315}}{20833331} x-\frac{25000}{20833331}=-\frac{17\sqrt{1081315}}{20833331}$$

Add \frac{25000}{20833331} to both sides of the equation.

$$x=\frac{17\sqrt{1081315}+25000}{20833331} x=\frac{25000-17\sqrt{1081315}}{20833331}$$