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Expression simplification/ Factor perfect squares/ -y^2+8*y*p+2*p^2

Factor -y^2+8*y*p+2*p^2 squared

An expression to simplify:

The solution

You have entered [src] 
   2              2
- y  + 8*y*p + 2*p
$$2 p^{2} + \left(p 8 y - y^{2}\right)$$ 
-y^2 + (8*y)*p + 2*p^2
Factorization [src] 
/      /         ___\\ /      /        ___\\
|    y*\-4 + 3*\/ 2 /| |    y*\4 + 3*\/ 2 /|
|p - ----------------|*|p + ---------------|
\           2        / \           2       /
$$\left(p - \frac{y \left(-4 + 3 \sqrt{2}\right)}{2}\right) \left(p + \frac{y \left(4 + 3 \sqrt{2}\right)}{2}\right)$$ 
(p - y*(-4 + 3*sqrt(2))/2)*(p + y*(4 + 3*sqrt(2))/2)
General simplification [src] 
   2      2        
- y  + 2*p  + 8*p*y
$$2 p^{2} + 8 p y - y^{2}$$ 
-y^2 + 2*p^2 + 8*p*y
The perfect square
Let's highlight the perfect square of the square three-member
$$2 p^{2} + \left(p 8 y - y^{2}\right)$$
Let us write down the identical expression
$$2 p^{2} + \left(p 8 y - y^{2}\right) = - 9 y^{2} + \left(2 p^{2} + 8 p y + 8 y^{2}\right)$$
or
$$2 p^{2} + \left(p 8 y - y^{2}\right) = - 9 y^{2} + \left(\sqrt{2} p + 2 \sqrt{2} y\right)^{2}$$
in the view of the product
$$\left(- \sqrt{9} y + \left(\sqrt{2} p + 2 \sqrt{2} y\right)\right) \left(\sqrt{9} y + \left(\sqrt{2} p + 2 \sqrt{2} y\right)\right)$$
$$\left(- 3 y + \left(\sqrt{2} p + 2 \sqrt{2} y\right)\right) \left(3 y + \left(\sqrt{2} p + 2 \sqrt{2} y\right)\right)$$
$$\left(\sqrt{2} p + y \left(-3 + 2 \sqrt{2}\right)\right) \left(\sqrt{2} p + y \left(2 \sqrt{2} + 3\right)\right)$$
$$\left(\sqrt{2} p + y \left(-3 + 2 \sqrt{2}\right)\right) \left(\sqrt{2} p + y \left(2 \sqrt{2} + 3\right)\right)$$
Numerical answer [src] 
-y^2 + 2.0*p^2 + 8.0*p*y
-y^2 + 2.0*p^2 + 8.0*p*y
Rational denominator [src] 
   2      2        
- y  + 2*p  + 8*p*y
$$2 p^{2} + 8 p y - y^{2}$$ 
-y^2 + 2*p^2 + 8*p*y
Assemble expression [src] 
   2      2        
- y  + 2*p  + 8*p*y
$$2 p^{2} + 8 p y - y^{2}$$ 
-y^2 + 2*p^2 + 8*p*y
Common denominator [src] 
   2      2        
- y  + 2*p  + 8*p*y
$$2 p^{2} + 8 p y - y^{2}$$ 
-y^2 + 2*p^2 + 8*p*y
Trigonometric part [src] 
   2      2        
- y  + 2*p  + 8*p*y
$$2 p^{2} + 8 p y - y^{2}$$ 
-y^2 + 2*p^2 + 8*p*y
Combinatorics [src] 
   2      2        
- y  + 2*p  + 8*p*y
$$2 p^{2} + 8 p y - y^{2}$$ 
-y^2 + 2*p^2 + 8*p*y
Combining rational expressions [src] 
   2               
2*p  + y*(-y + 8*p)
$$2 p^{2} + y \left(8 p - y\right)$$ 
2*p^2 + y*(-y + 8*p)
Powers [src] 
   2      2        
- y  + 2*p  + 8*p*y
$$2 p^{2} + 8 p y - y^{2}$$ 
-y^2 + 2*p^2 + 8*p*y
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