Solve 0=40b-b^2/2

$$0 = \frac { 40 b - b ^ { 2 } } { 2 }$$

Solve for b

$$b=40$$

$$b=0$$

Steps Using Factoring

Multiply both sides by 2. Anything times zero gives zero.

$$0=40b-b^{2}$$

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

$$40b-b^{2}=0$$

Factor out b.

$$b\left(40-b\right)=0$$

To find equation solutions, solve b=0 and 40-b=0.

$$b=0 b=40$$

Steps Using the Quadratic Formula

Multiply both sides by 2. Anything times zero gives zero.

$$0=40b-b^{2}$$

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

$$40b-b^{2}=0$$

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.

$$-b^{2}+40b=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 40 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-40±\sqrt{40^{2}}}{2\left(-1\right)}$$

Take the square root of 40^{2}.

$$b=\frac{-40±40}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{-40±40}{-2}$$

Now solve the equation b=\frac{-40±40}{-2} when ± is plus. Add -40 to 40.

$$b=\frac{0}{-2}$$

Divide 0 by -2.

$$b=0$$

Now solve the equation b=\frac{-40±40}{-2} when ± is minus. Subtract 40 from -40.

$$b=-\frac{80}{-2}$$

Divide -80 by -2.

$$b=40$$

The equation is now solved.

$$b=0 b=40$$

Steps for Completing the Square

Multiply both sides by 2. Anything times zero gives zero.

$$0=40b-b^{2}$$

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

$$40b-b^{2}=0$$

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.

$$-b^{2}+40b=0$$

Divide both sides by -1.

$$\frac{-b^{2}+40b}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$b^{2}+\frac{40}{-1}b=\frac{0}{-1}$$

Divide 40 by -1.

$$b^{2}-40b=\frac{0}{-1}$$

Divide 0 by -1.

$$b^{2}-40b=0$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-40b+\left(-20\right)^{2}=\left(-20\right)^{2}$$

Square -20.

$$b^{2}-40b+400=400$$

Factor b^{2}-40b+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-20\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-20\right)^{2}}=\sqrt{400}$$

Simplify.

$$b-20=20 b-20=-20$$

Add 20 to both sides of the equation.

$$b=40 b=0$$