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

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

An expression to simplify:

The solution

You have entered [src] 
   2            2
- y  + y*x + 2*x
$$2 x^{2} + \left(x y - y^{2}\right)$$ 
-y^2 + y*x + 2*x^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
$$2 x^{2} + \left(x y - y^{2}\right)$$
Let us write down the identical expression
$$2 x^{2} + \left(x y - y^{2}\right) = - \frac{9 y^{2}}{8} + \left(2 x^{2} + x y + \frac{y^{2}}{8}\right)$$
or
$$2 x^{2} + \left(x y - y^{2}\right) = - \frac{9 y^{2}}{8} + \left(\sqrt{2} x + \frac{\sqrt{2} y}{4}\right)^{2}$$
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)
Powers [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)
Numerical answer [src] 
-y^2 + 2.0*x^2 + x*y
-y^2 + 2.0*x^2 + x*y
Common denominator [src] 
   2      2      
- y  + 2*x  + x*y
$$2 x^{2} + x y - y^{2}$$ 
-y^2 + 2*x^2 + x*y
Combining rational expressions [src] 
   2            
2*x  + y*(x - y)
$$2 x^{2} + y \left(x - y\right)$$ 
2*x^2 + y*(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
Assemble expression [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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