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Expression simplification/ Factor perfect squares/ -y^2+y*x+5*x^2

Factor -y^2+y*x+5*x^2 squared

An expression to simplify:

The solution

You have entered [src] 
   2            2
- y  + y*x + 5*x
$$5 x^{2} + \left(x y - y^{2}\right)$$ 
-y^2 + y*x + 5*x^2
General simplification [src] 
   2      2      
- y  + 5*x  + x*y
$$5 x^{2} + x y - y^{2}$$ 
-y^2 + 5*x^2 + x*y
The perfect square
Let's highlight the perfect square of the square three-member
$$5 x^{2} + \left(x y - y^{2}\right)$$
Let us write down the identical expression
$$5 x^{2} + \left(x y - y^{2}\right) = - \frac{21 y^{2}}{20} + \left(5 x^{2} + x y + \frac{y^{2}}{20}\right)$$
or
$$5 x^{2} + \left(x y - y^{2}\right) = - \frac{21 y^{2}}{20} + \left(\sqrt{5} x + \frac{\sqrt{5} y}{10}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{21}{20}} y + \left(\sqrt{5} x + \frac{\sqrt{5}}{10} y\right)\right) \left(\sqrt{\frac{21}{20}} y + \left(\sqrt{5} x + \frac{\sqrt{5}}{10} y\right)\right)$$
$$\left(- \frac{\sqrt{105}}{10} y + \left(\sqrt{5} x + \frac{\sqrt{5}}{10} y\right)\right) \left(\frac{\sqrt{105}}{10} y + \left(\sqrt{5} x + \frac{\sqrt{5}}{10} y\right)\right)$$
$$\left(\sqrt{5} x + y \left(- \frac{\sqrt{105}}{10} + \frac{\sqrt{5}}{10}\right)\right) \left(\sqrt{5} x + y \left(\frac{\sqrt{5}}{10} + \frac{\sqrt{105}}{10}\right)\right)$$
$$\left(\sqrt{5} x + y \left(- \frac{\sqrt{105}}{10} + \frac{\sqrt{5}}{10}\right)\right) \left(\sqrt{5} x + y \left(\frac{\sqrt{5}}{10} + \frac{\sqrt{105}}{10}\right)\right)$$
Factorization [src] 
/      /       ____\\ /      /      ____\\
|    y*\-1 + \/ 21 /| |    y*\1 + \/ 21 /|
|x - ---------------|*|x + --------------|
\           10      / \          10      /
$$\left(x - \frac{y \left(-1 + \sqrt{21}\right)}{10}\right) \left(x + \frac{y \left(1 + \sqrt{21}\right)}{10}\right)$$ 
(x - y*(-1 + sqrt(21))/10)*(x + y*(1 + sqrt(21))/10)
Assemble expression [src] 
   2      2      
- y  + 5*x  + x*y
$$5 x^{2} + x y - y^{2}$$ 
-y^2 + 5*x^2 + x*y
Combinatorics [src] 
   2      2      
- y  + 5*x  + x*y
$$5 x^{2} + x y - y^{2}$$ 
-y^2 + 5*x^2 + x*y
Numerical answer [src] 
-y^2 + 5.0*x^2 + x*y
-y^2 + 5.0*x^2 + x*y
Combining rational expressions [src] 
   2            
5*x  + y*(x - y)
$$5 x^{2} + y \left(x - y\right)$$ 
5*x^2 + y*(x - y)
Trigonometric part [src] 
   2      2      
- y  + 5*x  + x*y
$$5 x^{2} + x y - y^{2}$$ 
-y^2 + 5*x^2 + x*y
Common denominator [src] 
   2      2      
- y  + 5*x  + x*y
$$5 x^{2} + x y - y^{2}$$ 
-y^2 + 5*x^2 + x*y
Rational denominator [src] 
   2      2      
- y  + 5*x  + x*y
$$5 x^{2} + x y - y^{2}$$ 
-y^2 + 5*x^2 + x*y
Powers [src] 
   2      2      
- y  + 5*x  + x*y
$$5 x^{2} + x y - y^{2}$$ 
-y^2 + 5*x^2 + x*y
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