Solve -100x^2+16000x+600000=40000

$$- 100 x ^ { 2 } + 16000 x + 600000 = 40000$$

Solve for x

$$x=20\sqrt{30}+80\approx 189.544511501$$

$$x=80-20\sqrt{30}\approx -29.544511501$$

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.

$$-100x^{2}+16000x+600000=40000$$

Subtract 40000 from both sides of the equation.

$$-100x^{2}+16000x+600000-40000=40000-40000$$

Subtracting 40000 from itself leaves 0.

$$-100x^{2}+16000x+600000-40000=0$$

Subtract 40000 from 600000.

$$-100x^{2}+16000x+560000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, 16000 for b, and 560000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16000±\sqrt{16000^{2}-4\left(-100\right)\times 560000}}{2\left(-100\right)}$$

Square 16000.

$$x=\frac{-16000±\sqrt{256000000-4\left(-100\right)\times 560000}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$x=\frac{-16000±\sqrt{256000000+400\times 560000}}{2\left(-100\right)}$$

Multiply 400 times 560000.

$$x=\frac{-16000±\sqrt{256000000+224000000}}{2\left(-100\right)}$$

Add 256000000 to 224000000.

$$x=\frac{-16000±\sqrt{480000000}}{2\left(-100\right)}$$

Take the square root of 480000000.

$$x=\frac{-16000±4000\sqrt{30}}{2\left(-100\right)}$$

Multiply 2 times -100.

$$x=\frac{-16000±4000\sqrt{30}}{-200}$$

Now solve the equation x=\frac{-16000±4000\sqrt{30}}{-200} when ± is plus. Add -16000 to 4000\sqrt{30}.

$$x=\frac{4000\sqrt{30}-16000}{-200}$$

Divide -16000+4000\sqrt{30} by -200.

$$x=80-20\sqrt{30}$$

Now solve the equation x=\frac{-16000±4000\sqrt{30}}{-200} when ± is minus. Subtract 4000\sqrt{30} from -16000.

$$x=\frac{-4000\sqrt{30}-16000}{-200}$$

Divide -16000-4000\sqrt{30} by -200.

$$x=20\sqrt{30}+80$$

The equation is now solved.

$$x=80-20\sqrt{30} x=20\sqrt{30}+80$$

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.

$$-100x^{2}+16000x+600000=40000$$

Subtract 600000 from both sides of the equation.

$$-100x^{2}+16000x+600000-600000=40000-600000$$

Subtracting 600000 from itself leaves 0.

$$-100x^{2}+16000x=40000-600000$$

Subtract 600000 from 40000.

$$-100x^{2}+16000x=-560000$$

Divide both sides by -100.

$$\frac{-100x^{2}+16000x}{-100}=-\frac{560000}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$x^{2}+\frac{16000}{-100}x=-\frac{560000}{-100}$$

Divide 16000 by -100.

$$x^{2}-160x=-\frac{560000}{-100}$$

Divide -560000 by -100.

$$x^{2}-160x=5600$$

Divide -160, the coefficient of the x term, by 2 to get -80. Then add the square of -80 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-160x+\left(-80\right)^{2}=5600+\left(-80\right)^{2}$$

Square -80.

$$x^{2}-160x+6400=5600+6400$$

Add 5600 to 6400.

$$x^{2}-160x+6400=12000$$

Factor x^{2}-160x+6400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-80\right)^{2}=12000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-80\right)^{2}}=\sqrt{12000}$$

Simplify.

$$x-80=20\sqrt{30} x-80=-20\sqrt{30}$$

Add 80 to both sides of the equation.

$$x=20\sqrt{30}+80 x=80-20\sqrt{30}$$