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Expression simplification/ Factor perfect squares/ y^4+y^2+2

Factor y^4+y^2+2 squared

An expression to simplify:

The solution

You have entered [src] 
 4    2    
y  + y  + 2
$$\left(y^{4} + y^{2}\right) + 2$$ 
y^4 + y^2 + 2
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + y^{2}\right) + 2$$
To do this, let's use the formula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 1$$
$$c = 2$$
Then
$$m = \frac{1}{2}$$
$$n = \frac{7}{4}$$
So,
$$\left(y^{2} + \frac{1}{2}\right)^{2} + \frac{7}{4}$$
Factorization [src] 
/             /    /  ___\\              /    /  ___\\\ /             /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\
|    4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /|| |    4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /|| |      4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /|| |      4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /||
|x + \/ 2 *sin|-----------| + I*\/ 2 *cos|-----------||*|x + \/ 2 *sin|-----------| - I*\/ 2 *cos|-----------||*|x + - \/ 2 *sin|-----------| + I*\/ 2 *cos|-----------||*|x + - \/ 2 *sin|-----------| - I*\/ 2 *cos|-----------||
\             \     2     /              \     2     // \             \     2     /              \     2     // \               \     2     /              \     2     // \               \     2     /              \     2     //
$$\left(x + \left(\sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right)$$ 
(((x + 2^(1/4)*sin(atan(sqrt(7))/2) + i*2^(1/4)*cos(atan(sqrt(7))/2))*(x + 2^(1/4)*sin(atan(sqrt(7))/2) - i*2^(1/4)*cos(atan(sqrt(7))/2)))*(x - 2^(1/4)*sin(atan(sqrt(7))/2) + i*2^(1/4)*cos(atan(sqrt(7))/2)))*(x - 2^(1/4)*sin(atan(sqrt(7))/2) - i*2^(1/4)*cos(atan(sqrt(7))/2))
General simplification [src] 
     2    4
2 + y  + y
$$y^{4} + y^{2} + 2$$ 
2 + y^2 + y^4
Rational denominator [src] 
     2    4
2 + y  + y
$$y^{4} + y^{2} + 2$$ 
2 + y^2 + y^4
Trigonometric part [src] 
     2    4
2 + y  + y
$$y^{4} + y^{2} + 2$$ 
2 + y^2 + y^4
Assemble expression [src] 
     2    4
2 + y  + y
$$y^{4} + y^{2} + 2$$ 
2 + y^2 + y^4
Powers [src] 
     2    4
2 + y  + y
$$y^{4} + y^{2} + 2$$ 
2 + y^2 + y^4
Common denominator [src] 
     2    4
2 + y  + y
$$y^{4} + y^{2} + 2$$ 
2 + y^2 + y^4
Combinatorics [src] 
     2    4
2 + y  + y
$$y^{4} + y^{2} + 2$$ 
2 + y^2 + y^4
Numerical answer [src] 
2.0 + y^2 + y^4
2.0 + y^2 + y^4
Combining rational expressions [src] 
     2 /     2\
2 + y *\1 + y /
$$y^{2} \left(y^{2} + 1\right) + 2$$ 
2 + y^2*(1 + y^2)
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