Solve v_s^2+30000v_{s}-900=0

$$v _ { s } ^ { 2 } + 30000 v _ { s } - 900 = 0$$

Solve for v_s

$$v_{s}=1590\sqrt{89}-15000\approx 0.02999997$$

$$v_{s}=-1590\sqrt{89}-15000\approx -30000.02999997$$

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.

$$v_{s}^{2}+30000v_{s}-900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30000 for b, and -900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v_{s}=\frac{-30000±\sqrt{30000^{2}-4\left(-900\right)}}{2}$$

Square 30000.

$$v_{s}=\frac{-30000±\sqrt{900000000-4\left(-900\right)}}{2}$$

Multiply -4 times -900.

$$v_{s}=\frac{-30000±\sqrt{900000000+3600}}{2}$$

Add 900000000 to 3600.

$$v_{s}=\frac{-30000±\sqrt{900003600}}{2}$$

Take the square root of 900003600.

$$v_{s}=\frac{-30000±3180\sqrt{89}}{2}$$

Now solve the equation v_{s}=\frac{-30000±3180\sqrt{89}}{2} when ± is plus. Add -30000 to 3180\sqrt{89}.

$$v_{s}=\frac{3180\sqrt{89}-30000}{2}$$

Divide -30000+3180\sqrt{89} by 2.

$$v_{s}=1590\sqrt{89}-15000$$

Now solve the equation v_{s}=\frac{-30000±3180\sqrt{89}}{2} when ± is minus. Subtract 3180\sqrt{89} from -30000.

$$v_{s}=\frac{-3180\sqrt{89}-30000}{2}$$

Divide -30000-3180\sqrt{89} by 2.

$$v_{s}=-1590\sqrt{89}-15000$$

The equation is now solved.

$$v_{s}=1590\sqrt{89}-15000 v_{s}=-1590\sqrt{89}-15000$$

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.

$$v_{s}^{2}+30000v_{s}-900=0$$

Add 900 to both sides of the equation.

$$v_{s}^{2}+30000v_{s}-900-\left(-900\right)=-\left(-900\right)$$

Subtracting -900 from itself leaves 0.

$$v_{s}^{2}+30000v_{s}=-\left(-900\right)$$

Subtract -900 from 0.

$$v_{s}^{2}+30000v_{s}=900$$

Divide 30000, the coefficient of the x term, by 2 to get 15000. Then add the square of 15000 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v_{s}^{2}+30000v_{s}+15000^{2}=900+15000^{2}$$

Square 15000.

$$v_{s}^{2}+30000v_{s}[phone]=900+225000000$$

Add 900 to 225000000.

$$v_{s}^{2}+30000v_{s}[phone]=225000900$$

Factor v_{s}^{2}+30000v_{s}[phone]. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v_{s}+15000\right)^{2}=225000900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v_{s}+15000\right)^{2}}=\sqrt{225000900}$$

Simplify.

$$v_{s}+15000=1590\sqrt{89} v_{s}+15000=-1590\sqrt{89}$$

Subtract 15000 from both sides of the equation.

$$v_{s}=1590\sqrt{89}-15000 v_{s}=-1590\sqrt{89}-15000$$