Solve 30000=-5x^2+1000x-5000

$$30000 = - 5 x ^ { 2 } + 1000 x - 5000$$

Solve for x

$$x=10\sqrt{30}+100\approx 154.772255751$$

$$x=100-10\sqrt{30}\approx 45.227744249$$

Steps Using the Quadratic Formula

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

$$-5x^{2}+1000x-5000=30000$$

Subtract 30000 from both sides.

$$-5x^{2}+1000x-5000-30000=0$$

Subtract 30000 from -5000 to get -35000.

$$-5x^{2}+1000x-35000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 1000 for b, and -35000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1000±\sqrt{1000^{2}-4\left(-5\right)\left(-35000\right)}}{2\left(-5\right)}$$

Square 1000.

$$x=\frac{-1000±\sqrt{1000000-4\left(-5\right)\left(-35000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-1000±\sqrt{1000000+20\left(-35000\right)}}{2\left(-5\right)}$$

Multiply 20 times -35000.

$$x=\frac{-1000±\sqrt{1000000-700000}}{2\left(-5\right)}$$

Add 1000000 to -700000.

$$x=\frac{-1000±\sqrt{300000}}{2\left(-5\right)}$$

Take the square root of 300000.

$$x=\frac{-1000±100\sqrt{30}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-1000±100\sqrt{30}}{-10}$$

Now solve the equation x=\frac{-1000±100\sqrt{30}}{-10} when ± is plus. Add -1000 to 100\sqrt{30}.

$$x=\frac{100\sqrt{30}-1000}{-10}$$

Divide -1000+100\sqrt{30} by -10.

$$x=100-10\sqrt{30}$$

Now solve the equation x=\frac{-1000±100\sqrt{30}}{-10} when ± is minus. Subtract 100\sqrt{30} from -1000.

$$x=\frac{-100\sqrt{30}-1000}{-10}$$

Divide -1000-100\sqrt{30} by -10.

$$x=10\sqrt{30}+100$$

The equation is now solved.

$$x=100-10\sqrt{30} x=10\sqrt{30}+100$$

Steps for Completing the Square

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

$$-5x^{2}+1000x-5000=30000$$

Add 5000 to both sides.

$$-5x^{2}+1000x=30000+5000$$

Add 30000 and 5000 to get 35000.

$$-5x^{2}+1000x=35000$$

Divide both sides by -5.

$$\frac{-5x^{2}+1000x}{-5}=\frac{35000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{1000}{-5}x=\frac{35000}{-5}$$

Divide 1000 by -5.

$$x^{2}-200x=\frac{35000}{-5}$$

Divide 35000 by -5.

$$x^{2}-200x=-7000$$

Divide -200, the coefficient of the x term, by 2 to get -100. Then add the square of -100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-200x+\left(-100\right)^{2}=-7000+\left(-100\right)^{2}$$

Square -100.

$$x^{2}-200x+10000=-7000+10000$$

Add -7000 to 10000.

$$x^{2}-200x+10000=3000$$

Factor x^{2}-200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-100\right)^{2}=3000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-100\right)^{2}}=\sqrt{3000}$$

Simplify.

$$x-100=10\sqrt{30} x-100=-10\sqrt{30}$$

Add 100 to both sides of the equation.

$$x=10\sqrt{30}+100 x=100-10\sqrt{30}$$