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

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

An expression to simplify:

The solution

You have entered [src] 
   2          2
2*x  + x*y - y
$$- y^{2} + \left(2 x^{2} + x y\right)$$ 
2*x^2 + x*y - y^2
General simplification [src] 
   2      2      
- y  + 2*x  + x*y
$$2 x^{2} + x y - y^{2}$$ 
-y^2 + 2*x^2 + x*y
The perfect square
Let's highlight the perfect square of the square three-member
$$- y^{2} + \left(2 x^{2} + x y\right)$$
Let us write down the identical expression
$$- y^{2} + \left(2 x^{2} + x y\right) = - \frac{9 y^{2}}{8} + \left(2 x^{2} + x y + \frac{y^{2}}{8}\right)$$
or
$$- y^{2} + \left(2 x^{2} + x y\right) = - \frac{9 y^{2}}{8} + \left(\sqrt{2} x + \frac{\sqrt{2} y}{4}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{9}{8}} y + \left(\sqrt{2} x + \frac{\sqrt{2}}{4} y\right)\right) \left(\sqrt{\frac{9}{8}} y + \left(\sqrt{2} x + \frac{\sqrt{2}}{4} y\right)\right)$$
$$\left(- \frac{3 \sqrt{2}}{4} y + \left(\sqrt{2} x + \frac{\sqrt{2}}{4} y\right)\right) \left(\frac{3 \sqrt{2}}{4} y + \left(\sqrt{2} x + \frac{\sqrt{2}}{4} y\right)\right)$$
$$\left(\sqrt{2} x + y \left(- \frac{3 \sqrt{2}}{4} + \frac{\sqrt{2}}{4}\right)\right) \left(\sqrt{2} x + y \left(\frac{\sqrt{2}}{4} + \frac{3 \sqrt{2}}{4}\right)\right)$$
$$\left(\sqrt{2} x - \frac{\sqrt{2} y}{2}\right) \left(\sqrt{2} x + \sqrt{2} y\right)$$
Factorization [src] 
        /    y\
(x + y)*|x - -|
        \    2/
$$\left(x - \frac{y}{2}\right) \left(x + y\right)$$ 
(x + y)*(x - y/2)
Assemble expression [src] 
   2      2      
- y  + 2*x  + x*y
$$2 x^{2} + x y - y^{2}$$ 
-y^2 + 2*x^2 + x*y
Rational denominator [src] 
   2      2      
- y  + 2*x  + x*y
$$2 x^{2} + x y - y^{2}$$ 
-y^2 + 2*x^2 + x*y
Combinatorics [src] 
(x + y)*(-y + 2*x)
$$\left(x + y\right) \left(2 x - y\right)$$ 
(x + y)*(-y + 2*x)
Common denominator [src] 
   2      2      
- y  + 2*x  + x*y
$$2 x^{2} + x y - y^{2}$$ 
-y^2 + 2*x^2 + x*y
Numerical answer [src] 
-y^2 + 2.0*x^2 + x*y
-y^2 + 2.0*x^2 + x*y
Combining rational expressions [src] 
   2              
- y  + x*(y + 2*x)
$$x \left(2 x + y\right) - y^{2}$$ 
-y^2 + x*(y + 2*x)
Powers [src] 
   2      2      
- y  + 2*x  + x*y
$$2 x^{2} + x y - y^{2}$$ 
-y^2 + 2*x^2 + x*y
Trigonometric part [src] 
   2      2      
- y  + 2*x  + x*y
$$2 x^{2} + x y - y^{2}$$ 
-y^2 + 2*x^2 + x*y
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