Solve -5000=-50x^2+1000x

$$- 5000 = - 50 x ^ { 2 } + 1000 x$$

Solve for x

$$x=10\sqrt{2}+10\approx 24.142135624$$

$$x=10-10\sqrt{2}\approx -4.142135624$$

Steps Using the Quadratic Formula

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

$$-50x^{2}+1000x=-5000$$

Add 5000 to both sides.

$$-50x^{2}+1000x+5000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50 for a, 1000 for b, and 5000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1000±\sqrt{1000^{2}-4\left(-50\right)\times 5000}}{2\left(-50\right)}$$

Square 1000.

$$x=\frac{-1000±\sqrt{1000000-4\left(-50\right)\times 5000}}{2\left(-50\right)}$$

Multiply -4 times -50.

$$x=\frac{-1000±\sqrt{1000000+200\times 5000}}{2\left(-50\right)}$$

Multiply 200 times 5000.

$$x=\frac{-1000±\sqrt{1000000+1000000}}{2\left(-50\right)}$$

Add 1000000 to 1000000.

$$x=\frac{-1000±\sqrt{2000000}}{2\left(-50\right)}$$

Take the square root of 2000000.

$$x=\frac{-1000±1000\sqrt{2}}{2\left(-50\right)}$$

Multiply 2 times -50.

$$x=\frac{-1000±1000\sqrt{2}}{-100}$$

Now solve the equation x=\frac{-1000±1000\sqrt{2}}{-100} when ± is plus. Add -1000 to 1000\sqrt{2}.

$$x=\frac{1000\sqrt{2}-1000}{-100}$$

Divide -1000+1000\sqrt{2} by -100.

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

Now solve the equation x=\frac{-1000±1000\sqrt{2}}{-100} when ± is minus. Subtract 1000\sqrt{2} from -1000.

$$x=\frac{-1000\sqrt{2}-1000}{-100}$$

Divide -1000-1000\sqrt{2} by -100.

$$x=10\sqrt{2}+10$$

The equation is now solved.

$$x=10-10\sqrt{2} x=10\sqrt{2}+10$$

Steps for Completing the Square

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

$$-50x^{2}+1000x=-5000$$

Divide both sides by -50.

$$\frac{-50x^{2}+1000x}{-50}=-\frac{5000}{-50}$$

Dividing by -50 undoes the multiplication by -50.

$$x^{2}+\frac{1000}{-50}x=-\frac{5000}{-50}$$

Divide 1000 by -50.

$$x^{2}-20x=-\frac{5000}{-50}$$

Divide -5000 by -50.

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

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.

$$x^{2}-20x+\left(-10\right)^{2}=100+\left(-10\right)^{2}$$

Square -10.

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

Add 100 to 100.

$$x^{2}-20x+100=200$$

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}.

$$\left(x-10\right)^{2}=200$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{200}$$

Simplify.

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

Add 10 to both sides of the equation.

$$x=10\sqrt{2}+10 x=10-10\sqrt{2}$$