Solve 80x=40x^2

$$80 x = 40 x ^ { 2 }$$

Solve for x

$$x=2$$

$$x=0$$

Steps Using Factoring

Subtract 40x^{2} from both sides.

$$80x-40x^{2}=0$$

Factor out x.

$$x\left(80-40x\right)=0$$

To find equation solutions, solve x=0 and 80-40x=0.

$$x=0 x=2$$

Steps Using the Quadratic Formula

Subtract 40x^{2} from both sides.

$$80x-40x^{2}=0$$

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.

$$-40x^{2}+80x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -40 for a, 80 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}}}{2\left(-40\right)}$$

Take the square root of 80^{2}.

$$x=\frac{-80±80}{2\left(-40\right)}$$

Multiply 2 times -40.

$$x=\frac{-80±80}{-80}$$

Now solve the equation x=\frac{-80±80}{-80} when ± is plus. Add -80 to 80.

$$x=\frac{0}{-80}$$

Divide 0 by -80.

$$x=0$$

Now solve the equation x=\frac{-80±80}{-80} when ± is minus. Subtract 80 from -80.

$$x=-\frac{160}{-80}$$

Divide -160 by -80.

$$x=2$$

The equation is now solved.

$$x=0 x=2$$

Steps for Completing the Square

Subtract 40x^{2} from both sides.

$$80x-40x^{2}=0$$

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.

$$-40x^{2}+80x=0$$

Divide both sides by -40.

$$\frac{-40x^{2}+80x}{-40}=\frac{0}{-40}$$

Dividing by -40 undoes the multiplication by -40.

$$x^{2}+\frac{80}{-40}x=\frac{0}{-40}$$

Divide 80 by -40.

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

Divide 0 by -40.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=1$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-1=1 x-1=-1$$

Add 1 to both sides of the equation.

$$x=2 x=0$$