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Expression simplification/ Factor perfect squares/ y^2-4*y*x+3*x^2

Factor y^2-4*y*x+3*x^2 squared

An expression to simplify:

The solution

You have entered [src] 
 2              2
y  - 4*y*x + 3*x
$$3 x^{2} + \left(- x 4 y + y^{2}\right)$$ 
y^2 - 4*y*x + 3*x^2
Factorization [src] 
/    y\        
|x - -|*(x - y)
\    3/
$$\left(x - y\right) \left(x - \frac{y}{3}\right)$$ 
(x - y/3)*(x - y)
General simplification [src] 
 2      2        
y  + 3*x  - 4*x*y
$$3 x^{2} - 4 x y + y^{2}$$ 
y^2 + 3*x^2 - 4*x*y
The perfect square
Let's highlight the perfect square of the square three-member
$$3 x^{2} + \left(- x 4 y + y^{2}\right)$$
Let us write down the identical expression
$$3 x^{2} + \left(- x 4 y + y^{2}\right) = - \frac{y^{2}}{3} + \left(3 x^{2} - 4 x y + \frac{4 y^{2}}{3}\right)$$
or
$$3 x^{2} + \left(- x 4 y + y^{2}\right) = - \frac{y^{2}}{3} + \left(\sqrt{3} x - \frac{2 \sqrt{3} y}{3}\right)^{2}$$
in the view of the product
$$\left(- \frac{y}{\sqrt{3}} + \left(\sqrt{3} x + - \frac{2 \sqrt{3}}{3} y\right)\right) \left(\frac{y}{\sqrt{3}} + \left(\sqrt{3} x + - \frac{2 \sqrt{3}}{3} y\right)\right)$$
$$\left(- \frac{\sqrt{3}}{3} y + \left(\sqrt{3} x + - \frac{2 \sqrt{3}}{3} y\right)\right) \left(\frac{\sqrt{3}}{3} y + \left(\sqrt{3} x + - \frac{2 \sqrt{3}}{3} y\right)\right)$$
$$\left(\sqrt{3} x + y \left(- \frac{2 \sqrt{3}}{3} - \frac{\sqrt{3}}{3}\right)\right) \left(\sqrt{3} x + y \left(- \frac{2 \sqrt{3}}{3} + \frac{\sqrt{3}}{3}\right)\right)$$
$$\left(\sqrt{3} x - \sqrt{3} y\right) \left(\sqrt{3} x - \frac{\sqrt{3} y}{3}\right)$$
Numerical answer [src] 
y^2 + 3.0*x^2 - 4.0*x*y
y^2 + 3.0*x^2 - 4.0*x*y
Rational denominator [src] 
 2      2        
y  + 3*x  - 4*x*y
$$3 x^{2} - 4 x y + y^{2}$$ 
y^2 + 3*x^2 - 4*x*y
Trigonometric part [src] 
 2      2        
y  + 3*x  - 4*x*y
$$3 x^{2} - 4 x y + y^{2}$$ 
y^2 + 3*x^2 - 4*x*y
Common denominator [src] 
 2      2        
y  + 3*x  - 4*x*y
$$3 x^{2} - 4 x y + y^{2}$$ 
y^2 + 3*x^2 - 4*x*y
Assemble expression [src] 
 2      2        
y  + 3*x  - 4*x*y
$$3 x^{2} - 4 x y + y^{2}$$ 
y^2 + 3*x^2 - 4*x*y
Powers [src] 
 2      2        
y  + 3*x  - 4*x*y
$$3 x^{2} - 4 x y + y^{2}$$ 
y^2 + 3*x^2 - 4*x*y
Combining rational expressions [src] 
   2              
3*x  + y*(y - 4*x)
$$3 x^{2} + y \left(- 4 x + y\right)$$ 
3*x^2 + y*(y - 4*x)
Combinatorics [src] 
(x - y)*(-y + 3*x)
$$\left(x - y\right) \left(3 x - y\right)$$ 
(x - y)*(-y + 3*x)
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