Solve (20-x)x=75

$$(20-x)x=75$$

Solve for x

$$x=5$$

$$x=15$$

Steps Using the Quadratic Formula

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

$$20x-x^{2}=75$$

Subtract 75 from both sides.

$$20x-x^{2}-75=0$$

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.

$$-x^{2}+20x-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 20 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-1\right)\left(-75\right)}}{2\left(-1\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-1\right)\left(-75\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-20±\sqrt{400+4\left(-75\right)}}{2\left(-1\right)}$$

Multiply 4 times -75.

$$x=\frac{-20±\sqrt{400-300}}{2\left(-1\right)}$$

Add 400 to -300.

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

Take the square root of 100.

$$x=\frac{-20±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-20±10}{-2}$$

Now solve the equation x=\frac{-20±10}{-2} when ± is plus. Add -20 to 10.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-20±10}{-2} when ± is minus. Subtract 10 from -20.

$$x=-\frac{30}{-2}$$

Divide -30 by -2.

$$x=15$$

The equation is now solved.

$$x=5 x=15$$

Steps for Completing the Square

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

$$20x-x^{2}=75$$

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.

$$-x^{2}+20x=75$$

Divide both sides by -1.

$$\frac{-x^{2}+20x}{-1}=\frac{75}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{20}{-1}x=\frac{75}{-1}$$

Divide 20 by -1.

$$x^{2}-20x=\frac{75}{-1}$$

Divide 75 by -1.

$$x^{2}-20x=-75$$

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.

$$x^{2}-20x+\left(-10\right)^{2}=-75+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-75+100$$

Add -75 to 100.

$$x^{2}-20x+100=25$$

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}.

$$\left(x-10\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-10=5 x-10=-5$$

Add 10 to both sides of the equation.

$$x=15 x=5$$