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Expression simplification/ Factor perfect squares/ -y^2+9*y*t+3*t^2

Factor -y^2+9*y*t+3*t^2 squared

An expression to simplify:

The solution

You have entered [src] 
   2              2
- y  + 9*y*t + 3*t
$$3 t^{2} + \left(t 9 y - y^{2}\right)$$ 
-y^2 + (9*y)*t + 3*t^2
Factorization [src] 
/      /       ____\\ /      /      ____\\
|    y*\-9 + \/ 93 /| |    y*\9 + \/ 93 /|
|t - ---------------|*|t + --------------|
\           6       / \          6       /
$$\left(t - \frac{y \left(-9 + \sqrt{93}\right)}{6}\right) \left(t + \frac{y \left(9 + \sqrt{93}\right)}{6}\right)$$ 
(t - y*(-9 + sqrt(93))/6)*(t + y*(9 + sqrt(93))/6)
The perfect square
Let's highlight the perfect square of the square three-member
$$3 t^{2} + \left(t 9 y - y^{2}\right)$$
Let us write down the identical expression
$$3 t^{2} + \left(t 9 y - y^{2}\right) = - \frac{31 y^{2}}{4} + \left(3 t^{2} + 9 t y + \frac{27 y^{2}}{4}\right)$$
or
$$3 t^{2} + \left(t 9 y - y^{2}\right) = - \frac{31 y^{2}}{4} + \left(\sqrt{3} t + \frac{3 \sqrt{3} y}{2}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{31}{4}} y + \left(\sqrt{3} t + \frac{3 \sqrt{3}}{2} y\right)\right) \left(\sqrt{\frac{31}{4}} y + \left(\sqrt{3} t + \frac{3 \sqrt{3}}{2} y\right)\right)$$
$$\left(- \frac{\sqrt{31}}{2} y + \left(\sqrt{3} t + \frac{3 \sqrt{3}}{2} y\right)\right) \left(\frac{\sqrt{31}}{2} y + \left(\sqrt{3} t + \frac{3 \sqrt{3}}{2} y\right)\right)$$
$$\left(\sqrt{3} t + y \left(\frac{3 \sqrt{3}}{2} + \frac{\sqrt{31}}{2}\right)\right) \left(\sqrt{3} t + y \left(- \frac{\sqrt{31}}{2} + \frac{3 \sqrt{3}}{2}\right)\right)$$
$$\left(\sqrt{3} t + y \left(\frac{3 \sqrt{3}}{2} + \frac{\sqrt{31}}{2}\right)\right) \left(\sqrt{3} t + y \left(- \frac{\sqrt{31}}{2} + \frac{3 \sqrt{3}}{2}\right)\right)$$
General simplification [src] 
   2      2        
- y  + 3*t  + 9*t*y
$$3 t^{2} + 9 t y - y^{2}$$ 
-y^2 + 3*t^2 + 9*t*y
Assemble expression [src] 
   2      2        
- y  + 3*t  + 9*t*y
$$3 t^{2} + 9 t y - y^{2}$$ 
-y^2 + 3*t^2 + 9*t*y
Combinatorics [src] 
   2      2        
- y  + 3*t  + 9*t*y
$$3 t^{2} + 9 t y - y^{2}$$ 
-y^2 + 3*t^2 + 9*t*y
Powers [src] 
   2      2        
- y  + 3*t  + 9*t*y
$$3 t^{2} + 9 t y - y^{2}$$ 
-y^2 + 3*t^2 + 9*t*y
Trigonometric part [src] 
   2      2        
- y  + 3*t  + 9*t*y
$$3 t^{2} + 9 t y - y^{2}$$ 
-y^2 + 3*t^2 + 9*t*y
Common denominator [src] 
   2      2        
- y  + 3*t  + 9*t*y
$$3 t^{2} + 9 t y - y^{2}$$ 
-y^2 + 3*t^2 + 9*t*y
Rational denominator [src] 
   2      2        
- y  + 3*t  + 9*t*y
$$3 t^{2} + 9 t y - y^{2}$$ 
-y^2 + 3*t^2 + 9*t*y
Combining rational expressions [src] 
   2               
3*t  + y*(-y + 9*t)
$$3 t^{2} + y \left(9 t - y\right)$$ 
3*t^2 + y*(-y + 9*t)
Numerical answer [src] 
-y^2 + 3.0*t^2 + 9.0*t*y
-y^2 + 3.0*t^2 + 9.0*t*y
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