Solve 100=x(20-x)

$$100=x(20-x)$$

Solve for x

$$x=10$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 20-x.

$$100=20x-x^{2}$$

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

$$20x-x^{2}=100$$

Subtract 100 from both sides.

$$20x-x^{2}-100=0$$

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.

$$-x^{2}+20x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 20 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-1\right)\left(-100\right)}}{2\left(-1\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-1\right)\left(-100\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-20±\sqrt{400+4\left(-100\right)}}{2\left(-1\right)}$$

Multiply 4 times -100.

$$x=\frac{-20±\sqrt{400-400}}{2\left(-1\right)}$$

Add 400 to -400.

$$x=\frac{-20±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=-\frac{20}{-2}$$

Divide -20 by -2.

$$x=10$$

Steps for Completing the Square

Use the distributive property to multiply x by 20-x.

$$100=20x-x^{2}$$

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

$$20x-x^{2}=100$$

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.

$$-x^{2}+20x=100$$

Divide both sides by -1.

$$\frac{-x^{2}+20x}{-1}=\frac{100}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{20}{-1}x=\frac{100}{-1}$$

Divide 20 by -1.

$$x^{2}-20x=\frac{100}{-1}$$

Divide 100 by -1.

$$x^{2}-20x=-100$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-100+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-100+100$$

Add -100 to 100.

$$x^{2}-20x+100=0$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-10=0 x-10=0$$

Add 10 to both sides of the equation.

$$x=10 x=10$$

The equation is now solved. Solutions are the same.

$$x=10$$