Solve 1000=100x-2x^2

$$1000 = 100 x - 2 x ^ { 2 }$$

Solve for x

$$x=5\sqrt{5}+25\approx 36.180339887$$

$$x=25-5\sqrt{5}\approx 13.819660113$$

Steps Using the Quadratic Formula

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

$$100x-2x^{2}=1000$$

Subtract 1000 from both sides.

$$100x-2x^{2}-1000=0$$

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.

$$-2x^{2}+100x-1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 100 for b, and -1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-2\right)\left(-1000\right)}}{2\left(-2\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-2\right)\left(-1000\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-100±\sqrt{10000+8\left(-1000\right)}}{2\left(-2\right)}$$

Multiply 8 times -1000.

$$x=\frac{-100±\sqrt{10000-8000}}{2\left(-2\right)}$$

Add 10000 to -8000.

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

Take the square root of 2000.

$$x=\frac{-100±20\sqrt{5}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-100±20\sqrt{5}}{-4}$$

Now solve the equation x=\frac{-100±20\sqrt{5}}{-4} when ± is plus. Add -100 to 20\sqrt{5}.

$$x=\frac{20\sqrt{5}-100}{-4}$$

Divide -100+20\sqrt{5} by -4.

$$x=25-5\sqrt{5}$$

Now solve the equation x=\frac{-100±20\sqrt{5}}{-4} when ± is minus. Subtract 20\sqrt{5} from -100.

$$x=\frac{-20\sqrt{5}-100}{-4}$$

Divide -100-20\sqrt{5} by -4.

$$x=5\sqrt{5}+25$$

The equation is now solved.

$$x=25-5\sqrt{5} x=5\sqrt{5}+25$$

Steps for Completing the Square

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

$$100x-2x^{2}=1000$$

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.

$$-2x^{2}+100x=1000$$

Divide both sides by -2.

$$\frac{-2x^{2}+100x}{-2}=\frac{1000}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{100}{-2}x=\frac{1000}{-2}$$

Divide 100 by -2.

$$x^{2}-50x=\frac{1000}{-2}$$

Divide 1000 by -2.

$$x^{2}-50x=-500$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-500+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-500+625$$

Add -500 to 625.

$$x^{2}-50x+625=125$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=125$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{125}$$

Simplify.

$$x-25=5\sqrt{5} x-25=-5\sqrt{5}$$

Add 25 to both sides of the equation.

$$x=5\sqrt{5}+25 x=25-5\sqrt{5}$$