Solve (x)=212x-5000-x^2

$$( x ) = 212 x - 5000 - x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{24521} + 211}{2} \approx 183.795913048$$

$$x = \frac{211 - \sqrt{24521}}{2} \approx 27.204086952$$

Steps Using the Quadratic Formula

Subtract 212x from both sides.

$$x-212x=-5000-x^{2}$$

Combine x and -212x to get -211x.

$$-211x=-5000-x^{2}$$

Subtract -5000 from both sides.

$$-211x-\left(-5000\right)=-x^{2}$$

The opposite of -5000 is 5000.

$$-211x+5000=-x^{2}$$

Add x^{2} to both sides.

$$-211x+5000+x^{2}=0$$

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.

$$x^{2}-211x+5000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -211 for b, and 5000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-211\right)±\sqrt{\left(-211\right)^{2}-4\times 5000}}{2}$$

Square -211.

$$x=\frac{-\left(-211\right)±\sqrt{44521-4\times 5000}}{2}$$

Multiply -4 times 5000.

$$x=\frac{-\left(-211\right)±\sqrt{44521-20000}}{2}$$

Add 44521 to -20000.

$$x=\frac{-\left(-211\right)±\sqrt{24521}}{2}$$

The opposite of -211 is 211.

$$x=\frac{211±\sqrt{24521}}{2}$$

Now solve the equation x=\frac{211±\sqrt{24521}}{2} when ± is plus. Add 211 to \sqrt{24521}.

$$x=\frac{\sqrt{24521}+211}{2}$$

Now solve the equation x=\frac{211±\sqrt{24521}}{2} when ± is minus. Subtract \sqrt{24521} from 211.

$$x=\frac{211-\sqrt{24521}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{24521}+211}{2} x=\frac{211-\sqrt{24521}}{2}$$

Steps for Completing the Square

Subtract 212x from both sides.

$$x-212x=-5000-x^{2}$$

Combine x and -212x to get -211x.

$$-211x=-5000-x^{2}$$

Add x^{2} to both sides.

$$-211x+x^{2}=-5000$$

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.

$$x^{2}-211x=-5000$$

Divide -211, the coefficient of the x term, by 2 to get -\frac{211}{2}. Then add the square of -\frac{211}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-211x+\left(-\frac{211}{2}\right)^{2}=-5000+\left(-\frac{211}{2}\right)^{2}$$

Square -\frac{211}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-211x+\frac{44521}{4}=-5000+\frac{44521}{4}$$

Add -5000 to \frac{44521}{4}.

$$x^{2}-211x+\frac{44521}{4}=\frac{24521}{4}$$

Factor x^{2}-211x+\frac{44521}{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}.

$$\left(x-\frac{211}{2}\right)^{2}=\frac{24521}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{211}{2}\right)^{2}}=\sqrt{\frac{24521}{4}}$$

Simplify.

$$x-\frac{211}{2}=\frac{\sqrt{24521}}{2} x-\frac{211}{2}=-\frac{\sqrt{24521}}{2}$$

Add \frac{211}{2} to both sides of the equation.

$$x=\frac{\sqrt{24521}+211}{2} x=\frac{211-\sqrt{24521}}{2}$$