Solve -10x^2+4000x-30000=12000

$$- 10 x ^ { 2 } + 4000 x - 30000 = 12000$$

Solve for x

$$x=10\sqrt{358}+200\approx 389.208879284$$

$$x=200-10\sqrt{358}\approx 10.791120716$$

Steps Using the Quadratic Formula

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.

$$-10x^{2}+4000x-30000=12000$$

Subtract 12000 from both sides of the equation.

$$-10x^{2}+4000x-30000-12000=12000-12000$$

Subtracting 12000 from itself leaves 0.

$$-10x^{2}+4000x-30000-12000=0$$

Subtract 12000 from -30000.

$$-10x^{2}+4000x-42000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 4000 for b, and -42000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4000±\sqrt{4000^{2}-4\left(-10\right)\left(-42000\right)}}{2\left(-10\right)}$$

Square 4000.

$$x=\frac{-4000±\sqrt{16000000-4\left(-10\right)\left(-42000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-4000±\sqrt{16000000+40\left(-42000\right)}}{2\left(-10\right)}$$

Multiply 40 times -42000.

$$x=\frac{-4000±\sqrt{16000000-1680000}}{2\left(-10\right)}$$

Add 16000000 to -1680000.

$$x=\frac{-4000±\sqrt{14320000}}{2\left(-10\right)}$$

Take the square root of 14320000.

$$x=\frac{-4000±200\sqrt{358}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-4000±200\sqrt{358}}{-20}$$

Now solve the equation x=\frac{-4000±200\sqrt{358}}{-20} when ± is plus. Add -4000 to 200\sqrt{358}.

$$x=\frac{200\sqrt{358}-4000}{-20}$$

Divide -4000+200\sqrt{358} by -20.

$$x=200-10\sqrt{358}$$

Now solve the equation x=\frac{-4000±200\sqrt{358}}{-20} when ± is minus. Subtract 200\sqrt{358} from -4000.

$$x=\frac{-200\sqrt{358}-4000}{-20}$$

Divide -4000-200\sqrt{358} by -20.

$$x=10\sqrt{358}+200$$

The equation is now solved.

$$x=200-10\sqrt{358} x=10\sqrt{358}+200$$

Steps for Completing the Square

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.

$$-10x^{2}+4000x-30000=12000$$

Add 30000 to both sides of the equation.

$$-10x^{2}+4000x-30000-\left(-30000\right)=12000-\left(-30000\right)$$

Subtracting -30000 from itself leaves 0.

$$-10x^{2}+4000x=12000-\left(-30000\right)$$

Subtract -30000 from 12000.

$$-10x^{2}+4000x=42000$$

Divide both sides by -10.

$$\frac{-10x^{2}+4000x}{-10}=\frac{42000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{4000}{-10}x=\frac{42000}{-10}$$

Divide 4000 by -10.

$$x^{2}-400x=\frac{42000}{-10}$$

Divide 42000 by -10.

$$x^{2}-400x=-4200$$

Divide -400, the coefficient of the x term, by 2 to get -200. Then add the square of -200 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-400x+\left(-200\right)^{2}=-4200+\left(-200\right)^{2}$$

Square -200.

$$x^{2}-400x+40000=-4200+40000$$

Add -4200 to 40000.

$$x^{2}-400x+40000=35800$$

Factor x^{2}-400x+40000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-200\right)^{2}=35800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-200\right)^{2}}=\sqrt{35800}$$

Simplify.

$$x-200=10\sqrt{358} x-200=-10\sqrt{358}$$

Add 200 to both sides of the equation.

$$x=10\sqrt{358}+200 x=200-10\sqrt{358}$$