Solve S(x)=210x-5000=x^2

$$S ( x ) = 210 x - 5000 = x ^ { 2 }$$

Solve for x

$$x=5\sqrt{241}+105\approx 182.620873481$$

$$x=105-5\sqrt{241}\approx 27.379126519$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$210x-5000-x^{2}=0$$

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.

$$-x^{2}+210x-5000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 210 for b, and -5000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-210±\sqrt{210^{2}-4\left(-1\right)\left(-5000\right)}}{2\left(-1\right)}$$

Square 210.

$$x=\frac{-210±\sqrt{44100-4\left(-1\right)\left(-5000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-210±\sqrt{44100+4\left(-5000\right)}}{2\left(-1\right)}$$

Multiply 4 times -5000.

$$x=\frac{-210±\sqrt{44100-20000}}{2\left(-1\right)}$$

Add 44100 to -20000.

$$x=\frac{-210±\sqrt{24100}}{2\left(-1\right)}$$

Take the square root of 24100.

$$x=\frac{-210±10\sqrt{241}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-210±10\sqrt{241}}{-2}$$

Now solve the equation x=\frac{-210±10\sqrt{241}}{-2} when ± is plus. Add -210 to 10\sqrt{241}.

$$x=\frac{10\sqrt{241}-210}{-2}$$

Divide -210+10\sqrt{241} by -2.

$$x=105-5\sqrt{241}$$

Now solve the equation x=\frac{-210±10\sqrt{241}}{-2} when ± is minus. Subtract 10\sqrt{241} from -210.

$$x=\frac{-10\sqrt{241}-210}{-2}$$

Divide -210-10\sqrt{241} by -2.

$$x=5\sqrt{241}+105$$

The equation is now solved.

$$x=105-5\sqrt{241} x=5\sqrt{241}+105$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$210x-5000-x^{2}=0$$

Add 5000 to both sides. Anything plus zero gives itself.

$$210x-x^{2}=5000$$

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.

$$-x^{2}+210x=5000$$

Divide both sides by -1.

$$\frac{-x^{2}+210x}{-1}=\frac{5000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{210}{-1}x=\frac{5000}{-1}$$

Divide 210 by -1.

$$x^{2}-210x=\frac{5000}{-1}$$

Divide 5000 by -1.

$$x^{2}-210x=-5000$$

Divide -210, the coefficient of the x term, by 2 to get -105. Then add the square of -105 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-210x+\left(-105\right)^{2}=-5000+\left(-105\right)^{2}$$

Square -105.

$$x^{2}-210x+11025=-5000+11025$$

Add -5000 to 11025.

$$x^{2}-210x+11025=6025$$

Factor x^{2}-210x+11025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-105\right)^{2}=6025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-105\right)^{2}}=\sqrt{6025}$$

Simplify.

$$x-105=5\sqrt{241} x-105=-5\sqrt{241}$$

Add 105 to both sides of the equation.

$$x=5\sqrt{241}+105 x=105-5\sqrt{241}$$