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Expression simplification/ Factor perfect squares/ x^4+2*x^2+8

Factor x^4+2*x^2+8 squared

An expression to simplify:

The solution

You have entered [src] 
 4      2    
x  + 2*x  + 8
$$\left(x^{4} + 2 x^{2}\right) + 8$$ 
x^4 + 2*x^2 + 8
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{4} + 2 x^{2}\right) + 8$$
To do this, let's use the formula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 2$$
$$c = 8$$
Then
$$m = 1$$
$$n = 7$$
So,
$$\left(x^{2} + 1\right)^{2} + 7$$
Factorization [src] 
/            /    /  ___\\             /    /  ___\\\ /            /    /  ___\\             /    /  ___\\\ /              /    /  ___\\             /    /  ___\\\ /              /    /  ___\\             /    /  ___\\\
|     3/4    |atan\\/ 7 /|      3/4    |atan\\/ 7 /|| |     3/4    |atan\\/ 7 /|      3/4    |atan\\/ 7 /|| |       3/4    |atan\\/ 7 /|      3/4    |atan\\/ 7 /|| |       3/4    |atan\\/ 7 /|      3/4    |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     //
$$\left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right)$$ 
(((x + 2^(3/4)*sin(atan(sqrt(7))/2) + i*2^(3/4)*cos(atan(sqrt(7))/2))*(x + 2^(3/4)*sin(atan(sqrt(7))/2) - i*2^(3/4)*cos(atan(sqrt(7))/2)))*(x - 2^(3/4)*sin(atan(sqrt(7))/2) + i*2^(3/4)*cos(atan(sqrt(7))/2)))*(x - 2^(3/4)*sin(atan(sqrt(7))/2) - i*2^(3/4)*cos(atan(sqrt(7))/2))
General simplification [src] 
     4      2
8 + x  + 2*x
$$x^{4} + 2 x^{2} + 8$$ 
8 + x^4 + 2*x^2
Trigonometric part [src] 
     4      2
8 + x  + 2*x
$$x^{4} + 2 x^{2} + 8$$ 
8 + x^4 + 2*x^2
Assemble expression [src] 
     4      2
8 + x  + 2*x
$$x^{4} + 2 x^{2} + 8$$ 
8 + x^4 + 2*x^2
Combining rational expressions [src] 
     2 /     2\
8 + x *\2 + x /
$$x^{2} \left(x^{2} + 2\right) + 8$$ 
8 + x^2*(2 + x^2)
Rational denominator [src] 
     4      2
8 + x  + 2*x
$$x^{4} + 2 x^{2} + 8$$ 
8 + x^4 + 2*x^2
Powers [src] 
     4      2
8 + x  + 2*x
$$x^{4} + 2 x^{2} + 8$$ 
8 + x^4 + 2*x^2
Combinatorics [src] 
     4      2
8 + x  + 2*x
$$x^{4} + 2 x^{2} + 8$$ 
8 + x^4 + 2*x^2
Numerical answer [src] 
8.0 + x^4 + 2.0*x^2
8.0 + x^4 + 2.0*x^2
Common denominator [src] 
     4      2
8 + x  + 2*x
$$x^{4} + 2 x^{2} + 8$$ 
8 + x^4 + 2*x^2
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