Solve d=-0.02d^2+2d

$$d = - 0.02 d ^ { 2 } + 2 d$$

Solve for d

$$d=50$$

$$d=0$$

Steps Using Factoring

Add 0.02d^{2} to both sides.

$$d+0.02d^{2}=2d$$

Subtract 2d from both sides.

$$d+0.02d^{2}-2d=0$$

Combine d and -2d to get -d.

$$-d+0.02d^{2}=0$$

Factor out d.

$$d\left(-1+0.02d\right)=0$$

To find equation solutions, solve d=0 and -1+\frac{d}{50}=0.

$$d=0 d=50$$

Steps Using the Quadratic Formula

Add 0.02d^{2} to both sides.

$$d+0.02d^{2}=2d$$

Subtract 2d from both sides.

$$d+0.02d^{2}-2d=0$$

Combine d and -2d to get -d.

$$-d+0.02d^{2}=0$$

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.

$$0.02d^{2}-d=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.02 for a, -1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-1\right)±\sqrt{1}}{2\times 0.02}$$

Take the square root of 1.

$$d=\frac{-\left(-1\right)±1}{2\times 0.02}$$

The opposite of -1 is 1.

$$d=\frac{1±1}{2\times 0.02}$$

Multiply 2 times 0.02.

$$d=\frac{1±1}{0.04}$$

Now solve the equation d=\frac{1±1}{0.04} when ± is plus. Add 1 to 1.

$$d=\frac{2}{0.04}$$

Divide 2 by 0.04 by multiplying 2 by the reciprocal of 0.04.

$$d=50$$

Now solve the equation d=\frac{1±1}{0.04} when ± is minus. Subtract 1 from 1.

$$d=\frac{0}{0.04}$$

Divide 0 by 0.04 by multiplying 0 by the reciprocal of 0.04.

$$d=0$$

The equation is now solved.

$$d=50 d=0$$

Steps for Completing the Square

Add 0.02d^{2} to both sides.

$$d+0.02d^{2}=2d$$

Subtract 2d from both sides.

$$d+0.02d^{2}-2d=0$$

Combine d and -2d to get -d.

$$-d+0.02d^{2}=0$$

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.

$$0.02d^{2}-d=0$$

Multiply both sides by 50.

$$\frac{0.02d^{2}-d}{0.02}=\frac{0}{0.02}$$

Dividing by 0.02 undoes the multiplication by 0.02.

$$d^{2}+\left(-\frac{1}{0.02}\right)d=\frac{0}{0.02}$$

Divide -1 by 0.02 by multiplying -1 by the reciprocal of 0.02.

$$d^{2}-50d=\frac{0}{0.02}$$

Divide 0 by 0.02 by multiplying 0 by the reciprocal of 0.02.

$$d^{2}-50d=0$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-50d+\left(-25\right)^{2}=\left(-25\right)^{2}$$

Square -25.

$$d^{2}-50d+625=625$$

Factor d^{2}-50d+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d-25\right)^{2}=625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-25\right)^{2}}=\sqrt{625}$$

Simplify.

$$d-25=25 d-25=-25$$

Add 25 to both sides of the equation.

$$d=50 d=0$$