Solve (x(100-x)=500

$$( x ( 100 - x ) = 500$$

Solve for x

$$x=20\sqrt{5}+50\approx 94.72135955$$

$$x=50-20\sqrt{5}\approx 5.27864045$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 100-x.

$$100x-x^{2}=500$$

Subtract 500 from both sides.

$$100x-x^{2}-500=0$$

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.

$$-x^{2}+100x-500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 100 for b, and -500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-100±\sqrt{10000+4\left(-500\right)}}{2\left(-1\right)}$$

Multiply 4 times -500.

$$x=\frac{-100±\sqrt{10000-2000}}{2\left(-1\right)}$$

Add 10000 to -2000.

$$x=\frac{-100±\sqrt{8000}}{2\left(-1\right)}$$

Take the square root of 8000.

$$x=\frac{-100±40\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-100±40\sqrt{5}}{-2}$$

Now solve the equation x=\frac{-100±40\sqrt{5}}{-2} when ± is plus. Add -100 to 40\sqrt{5}.

$$x=\frac{40\sqrt{5}-100}{-2}$$

Divide -100+40\sqrt{5} by -2.

$$x=50-20\sqrt{5}$$

Now solve the equation x=\frac{-100±40\sqrt{5}}{-2} when ± is minus. Subtract 40\sqrt{5} from -100.

$$x=\frac{-40\sqrt{5}-100}{-2}$$

Divide -100-40\sqrt{5} by -2.

$$x=20\sqrt{5}+50$$

The equation is now solved.

$$x=50-20\sqrt{5} x=20\sqrt{5}+50$$

Steps for Completing the Square

Use the distributive property to multiply x by 100-x.

$$100x-x^{2}=500$$

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.

$$-x^{2}+100x=500$$

Divide both sides by -1.

$$\frac{-x^{2}+100x}{-1}=\frac{500}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{100}{-1}x=\frac{500}{-1}$$

Divide 100 by -1.

$$x^{2}-100x=\frac{500}{-1}$$

Divide 500 by -1.

$$x^{2}-100x=-500$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-500+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-500+2500$$

Add -500 to 2500.

$$x^{2}-100x+2500=2000$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=2000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2000}$$

Simplify.

$$x-50=20\sqrt{5} x-50=-20\sqrt{5}$$

Add 50 to both sides of the equation.

$$x=20\sqrt{5}+50 x=50-20\sqrt{5}$$