Solve 0.5x^2+0.5x=100x

$$0.5 { x }^{ 2 } +0.5x=100x$$

Solve for x

$$x=199$$

$$x=0$$

Steps Using Factoring

Subtract 100x from both sides.

$$0.5x^{2}+0.5x-100x=0$$

Combine 0.5x and -100x to get -99.5x.

$$0.5x^{2}-99.5x=0$$

Factor out x.

$$x\left(0.5x-99.5\right)=0$$

To find equation solutions, solve x=0 and \frac{x-199}{2}=0.

$$x=0 x=199$$

Steps Using the Quadratic Formula

Subtract 100x from both sides.

$$0.5x^{2}+0.5x-100x=0$$

Combine 0.5x and -100x to get -99.5x.

$$0.5x^{2}-99.5x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.5 for a, -99.5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-99.5\right)±\sqrt{\left(-99.5\right)^{2}}}{2\times 0.5}$$

Take the square root of \left(-99.5\right)^{2}.

$$x=\frac{-\left(-99.5\right)±\frac{199}{2}}{2\times 0.5}$$

The opposite of -99.5 is 99.5.

$$x=\frac{99.5±\frac{199}{2}}{2\times 0.5}$$

Multiply 2 times 0.5.

$$x=\frac{99.5±\frac{199}{2}}{1}$$

Now solve the equation x=\frac{99.5±\frac{199}{2}}{1} when ± is plus. Add 99.5 to \frac{199}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{199}{1}$$

Divide 199 by 1.

$$x=199$$

Now solve the equation x=\frac{99.5±\frac{199}{2}}{1} when ± is minus. Subtract \frac{199}{2} from 99.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{1}$$

Divide 0 by 1.

$$x=0$$

The equation is now solved.

$$x=199 x=0$$

Steps for Completing the Square

Subtract 100x from both sides.

$$0.5x^{2}+0.5x-100x=0$$

Combine 0.5x and -100x to get -99.5x.

$$0.5x^{2}-99.5x=0$$

Multiply both sides by 2.

$$\frac{0.5x^{2}-99.5x}{0.5}=\frac{0}{0.5}$$

Dividing by 0.5 undoes the multiplication by 0.5.

$$x^{2}+\left(-\frac{99.5}{0.5}\right)x=\frac{0}{0.5}$$

Divide -99.5 by 0.5 by multiplying -99.5 by the reciprocal of 0.5.

$$x^{2}-199x=\frac{0}{0.5}$$

Divide 0 by 0.5 by multiplying 0 by the reciprocal of 0.5.

$$x^{2}-199x=0$$

Divide -199, the coefficient of the x term, by 2 to get -\frac{199}{2}. Then add the square of -\frac{199}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-199x+\left(-\frac{199}{2}\right)^{2}=\left(-\frac{199}{2}\right)^{2}$$

Square -\frac{199}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-199x+9900.25=9900.25$$

Factor x^{2}-199x+9900.25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{199}{2}\right)^{2}=9900.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{199}{2}\right)^{2}}=\sqrt{9900.25}$$

Simplify.

$$x-\frac{199}{2}=\frac{199}{2} x-\frac{199}{2}=-\frac{199}{2}$$

Add \frac{199}{2} to both sides of the equation.

$$x=199 x=0$$