Solve 35000-200000x-500x^2

$$35000-200000x-500 { x }^{ 2 }$$

Factor

$$-500\left(x-\left(-\sqrt{40070}-200\right)\right)\left(x-\left(\sqrt{40070}-200\right)\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-500x^{2}-200000x+35000=0$$

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.

$$x=\frac{-\left(-200000\right)±\sqrt{\left(-200000\right)^{2}-4\left(-500\right)\times 35000}}{2\left(-500\right)}$$

Square -200000.

$$x=\frac{-\left(-200000\right)±\sqrt{40000000000-4\left(-500\right)\times 35000}}{2\left(-500\right)}$$

Multiply -4 times -500.

$$x=\frac{-\left(-200000\right)±\sqrt{40000000000+2000\times 35000}}{2\left(-500\right)}$$

Multiply 2000 times 35000.

$$x=\frac{-\left(-200000\right)±\sqrt{40000000000+70000000}}{2\left(-500\right)}$$

Add 40000000000 to 70000000.

$$x=\frac{-\left(-200000\right)±\sqrt{40070000000}}{2\left(-500\right)}$$

Take the square root of 40070000000.

$$x=\frac{-\left(-200000\right)±1000\sqrt{40070}}{2\left(-500\right)}$$

The opposite of -200000 is 200000.

$$x=\frac{200000±1000\sqrt{40070}}{2\left(-500\right)}$$

Multiply 2 times -500.

$$x=\frac{200000±1000\sqrt{40070}}{-1000}$$

Now solve the equation x=\frac{200000±1000\sqrt{40070}}{-1000} when ± is plus. Add 200000 to 1000\sqrt{40070}.

$$x=\frac{1000\sqrt{40070}+200000}{-1000}$$

Divide 200000+1000\sqrt{40070} by -1000.

$$x=-\left(\sqrt{40070}+200\right)$$

Now solve the equation x=\frac{200000±1000\sqrt{40070}}{-1000} when ± is minus. Subtract 1000\sqrt{40070} from 200000.

$$x=\frac{200000-1000\sqrt{40070}}{-1000}$$

Divide 200000-1000\sqrt{40070} by -1000.

$$x=\sqrt{40070}-200$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\left(200+\sqrt{40070}\right) for x_{1} and -200+\sqrt{40070} for x_{2}.

$$-500x^{2}-200000x+35000=-500\left(x-\left(-\left(\sqrt{40070}+200\right)\right)\right)\left(x-\left(\sqrt{40070}-200\right)\right)$$

Evaluate

$$35000-200000x-500x^{2}$$