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Expression simplification/ Factor perfect squares/ -y^2-4*y*x+6*x^2

Factor -y^2-4*y*x+6*x^2 squared

An expression to simplify:

The solution

You have entered [src] 
   2              2
- y  - 4*y*x + 6*x
$$6 x^{2} + \left(- x 4 y - y^{2}\right)$$ 
-y^2 - 4*y*x + 6*x^2
General simplification [src] 
   2      2        
- y  + 6*x  - 4*x*y
$$6 x^{2} - 4 x y - y^{2}$$ 
-y^2 + 6*x^2 - 4*x*y
The perfect square
Let's highlight the perfect square of the square three-member
$$6 x^{2} + \left(- x 4 y - y^{2}\right)$$
Let us write down the identical expression
$$6 x^{2} + \left(- x 4 y - y^{2}\right) = - \frac{5 y^{2}}{3} + \left(6 x^{2} - 4 x y + \frac{2 y^{2}}{3}\right)$$
or
$$6 x^{2} + \left(- x 4 y - y^{2}\right) = - \frac{5 y^{2}}{3} + \left(\sqrt{6} x - \frac{\sqrt{6} y}{3}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{5}{3}} y + \left(\sqrt{6} x + - \frac{\sqrt{6}}{3} y\right)\right) \left(\sqrt{\frac{5}{3}} y + \left(\sqrt{6} x + - \frac{\sqrt{6}}{3} y\right)\right)$$
$$\left(- \frac{\sqrt{15}}{3} y + \left(\sqrt{6} x + - \frac{\sqrt{6}}{3} y\right)\right) \left(\frac{\sqrt{15}}{3} y + \left(\sqrt{6} x + - \frac{\sqrt{6}}{3} y\right)\right)$$
$$\left(\sqrt{6} x + y \left(- \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3}\right)\right) \left(\sqrt{6} x + y \left(- \frac{\sqrt{6}}{3} + \frac{\sqrt{15}}{3}\right)\right)$$
$$\left(\sqrt{6} x + y \left(- \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3}\right)\right) \left(\sqrt{6} x + y \left(- \frac{\sqrt{6}}{3} + \frac{\sqrt{15}}{3}\right)\right)$$
Factorization [src] 
/      /      ____\\ /      /      ____\\
|    y*\2 - \/ 10 /| |    y*\2 + \/ 10 /|
|x - --------------|*|x - --------------|
\          6       / \          6       /
$$\left(x - \frac{y \left(2 - \sqrt{10}\right)}{6}\right) \left(x - \frac{y \left(2 + \sqrt{10}\right)}{6}\right)$$ 
(x - y*(2 - sqrt(10))/6)*(x - y*(2 + sqrt(10))/6)
Numerical answer [src] 
-y^2 + 6.0*x^2 - 4.0*x*y
-y^2 + 6.0*x^2 - 4.0*x*y
Common denominator [src] 
   2      2        
- y  + 6*x  - 4*x*y
$$6 x^{2} - 4 x y - y^{2}$$ 
-y^2 + 6*x^2 - 4*x*y
Powers [src] 
   2      2        
- y  + 6*x  - 4*x*y
$$6 x^{2} - 4 x y - y^{2}$$ 
-y^2 + 6*x^2 - 4*x*y
Combinatorics [src] 
   2      2        
- y  + 6*x  - 4*x*y
$$6 x^{2} - 4 x y - y^{2}$$ 
-y^2 + 6*x^2 - 4*x*y
Rational denominator [src] 
   2      2        
- y  + 6*x  - 4*x*y
$$6 x^{2} - 4 x y - y^{2}$$ 
-y^2 + 6*x^2 - 4*x*y
Combining rational expressions [src] 
   2               
6*x  + y*(-y - 4*x)
$$6 x^{2} + y \left(- 4 x - y\right)$$ 
6*x^2 + y*(-y - 4*x)
Trigonometric part [src] 
   2      2        
- y  + 6*x  - 4*x*y
$$6 x^{2} - 4 x y - y^{2}$$ 
-y^2 + 6*x^2 - 4*x*y
Assemble expression [src] 
   2      2        
- y  + 6*x  - 4*x*y
$$6 x^{2} - 4 x y - y^{2}$$ 
-y^2 + 6*x^2 - 4*x*y
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