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xy-y^2 equation

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Numerical solution:

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The solution

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       2    
x*y - y  = 0
xy−y2=0x y - y^{2} = 0
Simplify
Detail solution
This equation is of the form
a*y^2 + b*y + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
y1=D−b2ay_{1} = \frac{\sqrt{D} - b}{2 a}
y2=−D−b2ay_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=−1a = -1
b=xb = x
c=0c = 0
, then
D = b^2 - 4 * a * c = 

(x)^2 - 4 * (-1) * (0) = x^2

The equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)

y2 = (-b - sqrt(D)) / (2*a)

or
y1=x2−x22y_{1} = \frac{x}{2} - \frac{\sqrt{x^{2}}}{2}
y2=x2+x22y_{2} = \frac{x}{2} + \frac{\sqrt{x^{2}}}{2}
Vieta's Theorem
rewrite the equation
xy−y2=0x y - y^{2} = 0
of
ay2+by+c=0a y^{2} + b y + c = 0
as reduced quadratic equation
y2+bya+ca=0y^{2} + \frac{b y}{a} + \frac{c}{a} = 0
−xy+y2=0- x y + y^{2} = 0
py+q+y2=0p y + q + y^{2} = 0
where
p=bap = \frac{b}{a}
p=−xp = - x
q=caq = \frac{c}{a}
q=0q = 0
Vieta Formulas
y1+y2=−py_{1} + y_{2} = - p
y1y2=qy_{1} y_{2} = q
y1+y2=xy_{1} + y_{2} = x
y1y2=0y_{1} y_{2} = 0
The graph
Sum and product of roots [src] 
sum
I*im(x) + re(x)
re⁡(x)+iim⁡(x)\operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)} 
=
I*im(x) + re(x)
re⁡(x)+iim⁡(x)\operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)} 
product
0*(I*im(x) + re(x))
0(re⁡(x)+iim⁡(x))0 \left(\operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)}\right) 
=
0
00 
0
Rapid solution [src] 
y1 = 0
y1=0y_{1} = 0 
y2 = I*im(x) + re(x)
y2=re⁡(x)+iim⁡(x)y_{2} = \operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)} 
y2 = re(x) + i*im(x)
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