Solve 10=x(2-0.025x)

$$10=x(2-0.025x)$$

Solve for x

$$x=20\sqrt{3}+40\approx 74.641016151$$

$$x=40-20\sqrt{3}\approx 5.358983849$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 2-0.025x.

$$10=2x-0.025x^{2}$$

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

$$2x-0.025x^{2}=10$$

Subtract 10 from both sides.

$$2x-0.025x^{2}-10=0$$

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.

$$-0.025x^{2}+2x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.025 for a, 2 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-0.025\right)\left(-10\right)}}{2\left(-0.025\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-0.025\right)\left(-10\right)}}{2\left(-0.025\right)}$$

Multiply -4 times -0.025.

$$x=\frac{-2±\sqrt{4+0.1\left(-10\right)}}{2\left(-0.025\right)}$$

Multiply 0.1 times -10.

$$x=\frac{-2±\sqrt{4-1}}{2\left(-0.025\right)}$$

Add 4 to -1.

$$x=\frac{-2±\sqrt{3}}{2\left(-0.025\right)}$$

Multiply 2 times -0.025.

$$x=\frac{-2±\sqrt{3}}{-0.05}$$

Now solve the equation x=\frac{-2±\sqrt{3}}{-0.05} when ± is plus. Add -2 to \sqrt{3}.

$$x=\frac{\sqrt{3}-2}{-0.05}$$

Divide -2+\sqrt{3} by -0.05 by multiplying -2+\sqrt{3} by the reciprocal of -0.05.

$$x=40-20\sqrt{3}$$

Now solve the equation x=\frac{-2±\sqrt{3}}{-0.05} when ± is minus. Subtract \sqrt{3} from -2.

$$x=\frac{-\sqrt{3}-2}{-0.05}$$

Divide -2-\sqrt{3} by -0.05 by multiplying -2-\sqrt{3} by the reciprocal of -0.05.

$$x=20\sqrt{3}+40$$

The equation is now solved.

$$x=40-20\sqrt{3} x=20\sqrt{3}+40$$

Steps for Completing the Square

Use the distributive property to multiply x by 2-0.025x.

$$10=2x-0.025x^{2}$$

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

$$2x-0.025x^{2}=10$$

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.

$$-0.025x^{2}+2x=10$$

Multiply both sides by -40.

$$\frac{-0.025x^{2}+2x}{-0.025}=\frac{10}{-0.025}$$

Dividing by -0.025 undoes the multiplication by -0.025.

$$x^{2}+\frac{2}{-0.025}x=\frac{10}{-0.025}$$

Divide 2 by -0.025 by multiplying 2 by the reciprocal of -0.025.

$$x^{2}-80x=\frac{10}{-0.025}$$

Divide 10 by -0.025 by multiplying 10 by the reciprocal of -0.025.

$$x^{2}-80x=-400$$

Divide -80, the coefficient of the x term, by 2 to get -40. Then add the square of -40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-80x+\left(-40\right)^{2}=-400+\left(-40\right)^{2}$$

Square -40.

$$x^{2}-80x+1600=-400+1600$$

Add -400 to 1600.

$$x^{2}-80x+1600=1200$$

Factor x^{2}-80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-40\right)^{2}=1200$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{1200}$$

Simplify.

$$x-40=20\sqrt{3} x-40=-20\sqrt{3}$$

Add 40 to both sides of the equation.

$$x=20\sqrt{3}+40 x=40-20\sqrt{3}$$