Solve x=25000+150x-x^2

$$x=25000+150x- { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{122201} + 149}{2} \approx 249.28629809$$

$$x=\frac{149-\sqrt{122201}}{2}\approx -100.28629809$$

Steps Using the Quadratic Formula

Subtract 25000 from both sides.

$$x-25000=150x-x^{2}$$

Subtract 150x from both sides.

$$x-25000-150x=-x^{2}$$

Combine x and -150x to get -149x.

$$-149x-25000=-x^{2}$$

Add x^{2} to both sides.

$$-149x-25000+x^{2}=0$$

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.

$$x^{2}-149x-25000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -149 for b, and -25000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-149\right)±\sqrt{\left(-149\right)^{2}-4\left(-25000\right)}}{2}$$

Square -149.

$$x=\frac{-\left(-149\right)±\sqrt{22201-4\left(-25000\right)}}{2}$$

Multiply -4 times -25000.

$$x=\frac{-\left(-149\right)±\sqrt{22201+100000}}{2}$$

Add 22201 to 100000.

$$x=\frac{-\left(-149\right)±\sqrt{122201}}{2}$$

The opposite of -149 is 149.

$$x=\frac{149±\sqrt{122201}}{2}$$

Now solve the equation x=\frac{149±\sqrt{122201}}{2} when ± is plus. Add 149 to \sqrt{122201}.

$$x=\frac{\sqrt{122201}+149}{2}$$

Now solve the equation x=\frac{149±\sqrt{122201}}{2} when ± is minus. Subtract \sqrt{122201} from 149.

$$x=\frac{149-\sqrt{122201}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{122201}+149}{2} x=\frac{149-\sqrt{122201}}{2}$$

Steps for Completing the Square

Subtract 150x from both sides.

$$x-150x=25000-x^{2}$$

Combine x and -150x to get -149x.

$$-149x=25000-x^{2}$$

Add x^{2} to both sides.

$$-149x+x^{2}=25000$$

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.

$$x^{2}-149x=25000$$

Divide -149, the coefficient of the x term, by 2 to get -\frac{149}{2}. Then add the square of -\frac{149}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-149x+\left(-\frac{149}{2}\right)^{2}=25000+\left(-\frac{149}{2}\right)^{2}$$

Square -\frac{149}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-149x+\frac{22201}{4}=25000+\frac{22201}{4}$$

Add 25000 to \frac{22201}{4}.

$$x^{2}-149x+\frac{22201}{4}=\frac{122201}{4}$$

Factor x^{2}-149x+\frac{22201}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{149}{2}\right)^{2}=\frac{122201}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{149}{2}\right)^{2}}=\sqrt{\frac{122201}{4}}$$

Simplify.

$$x-\frac{149}{2}=\frac{\sqrt{122201}}{2} x-\frac{149}{2}=-\frac{\sqrt{122201}}{2}$$

Add \frac{149}{2} to both sides of the equation.

$$x=\frac{\sqrt{122201}+149}{2} x=\frac{149-\sqrt{122201}}{2}$$