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2000+5000/x-3000-2000/x-500/x^2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
5000 2000 500
2000 + ---- - 3000 - ---- - --- = 0
x x 2
x
$$\left(\left(\left(2000 + \frac{5000}{x}\right) - 3000\right) - \frac{2000}{x}\right) - \frac{500}{x^{2}} = 0$$
Detail solution
Given the equation:
$$\left(\left(\left(2000 + \frac{5000}{x}\right) - 3000\right) - \frac{2000}{x}\right) - \frac{500}{x^{2}} = 0$$
Multiply the equation sides by the denominators:
x^2
we get:
$$x^{2} \left(\left(\left(\left(2000 + \frac{5000}{x}\right) - 3000\right) - \frac{2000}{x}\right) - \frac{500}{x^{2}}\right) = 0$$
$$- 1000 x^{2} + 3000 x - 500 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -1000$$
$$b = 3000$$
$$c = -500$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{3}{2} - \frac{\sqrt{7}}{2}$$
$$x_{2} = \frac{\sqrt{7}}{2} + \frac{3}{2}$$
$$\left(\left(\left(2000 + \frac{5000}{x}\right) - 3000\right) - \frac{2000}{x}\right) - \frac{500}{x^{2}} = 0$$
Multiply the equation sides by the denominators:
x^2
we get:
$$x^{2} \left(\left(\left(\left(2000 + \frac{5000}{x}\right) - 3000\right) - \frac{2000}{x}\right) - \frac{500}{x^{2}}\right) = 0$$
$$- 1000 x^{2} + 3000 x - 500 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -1000$$
$$b = 3000$$
$$c = -500$$
, then
D = b^2 - 4 * a * c =
(3000)^2 - 4 * (-1000) * (-500) = 7000000
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{3}{2} - \frac{\sqrt{7}}{2}$$
$$x_{2} = \frac{\sqrt{7}}{2} + \frac{3}{2}$$
Rapid solution
[src]
___
3 \/ 7
x1 = - - -----
2 2
$$x_{1} = \frac{3}{2} - \frac{\sqrt{7}}{2}$$
___
3 \/ 7
x2 = - + -----
2 2
$$x_{2} = \frac{\sqrt{7}}{2} + \frac{3}{2}$$
x2 = sqrt(7)/2 + 3/2
Sum and product of roots
[src]
sum
___ ___ 3 \/ 7 3 \/ 7 - - ----- + - + ----- 2 2 2 2
$$\left(\frac{3}{2} - \frac{\sqrt{7}}{2}\right) + \left(\frac{\sqrt{7}}{2} + \frac{3}{2}\right)$$
=
3
$$3$$
product
/ ___\ / ___\ |3 \/ 7 | |3 \/ 7 | |- - -----|*|- + -----| \2 2 / \2 2 /
$$\left(\frac{3}{2} - \frac{\sqrt{7}}{2}\right) \left(\frac{\sqrt{7}}{2} + \frac{3}{2}\right)$$
=
1/2
$$\frac{1}{2}$$
1/2