Solve (100+x)(200-x)=20000

$$( 100 + x ) ( 200 - x ) = 20000$$

Solve for x

$$x=100$$

$$x=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100+x by 200-x and combine like terms.

$$20000+100x-x^{2}=20000$$

Subtract 20000 from both sides.

$$20000+100x-x^{2}-20000=0$$

Subtract 20000 from 20000 to get 0.

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

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.

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

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

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

Take the square root of 100^{2}.

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

Multiply 2 times -1.

$$x=\frac{-100±100}{-2}$$

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

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

Divide 0 by -2.

$$x=0$$

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

$$x=-\frac{200}{-2}$$

Divide -200 by -2.

$$x=100$$

The equation is now solved.

$$x=0 x=100$$

Steps for Completing the Square

Use the distributive property to multiply 100+x by 200-x and combine like terms.

$$20000+100x-x^{2}=20000$$

Subtract 20000 from both sides.

$$100x-x^{2}=20000-20000$$

Subtract 20000 from 20000 to get 0.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+100x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{100}{-1}x=\frac{0}{-1}$$

Divide 100 by -1.

$$x^{2}-100x=\frac{0}{-1}$$

Divide 0 by -1.

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

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.

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

Square -50.

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

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}.

$$\left(x-50\right)^{2}=2500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2500}$$

Simplify.

$$x-50=50 x-50=-50$$

Add 50 to both sides of the equation.

$$x=100 x=0$$