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

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

An expression to simplify:

The solution

You have entered [src] 
   4    2    
4*x  - x  + 2
$$\left(4 x^{4} - x^{2}\right) + 2$$ 
4*x^4 - x^2 + 2
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(4 x^{4} - x^{2}\right) + 2$$
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
$$a = 4$$
$$b = -1$$
$$c = 2$$
Then
$$m = - \frac{1}{8}$$
$$n = \frac{31}{16}$$
So,
$$4 \left(x^{2} - \frac{1}{8}\right)^{2} + \frac{31}{16}$$
Factorization [src] 
/            /    /  ____\\             /    /  ____\\\ /            /    /  ____\\             /    /  ____\\\ /              /    /  ____\\             /    /  ____\\\ /              /    /  ____\\             /    /  ____\\\
|     3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /|| |     3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /|| |       3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /|| |       3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /||
|    2   *cos|------------|   I*2   *sin|------------|| |    2   *cos|------------|   I*2   *sin|------------|| |      2   *cos|------------|   I*2   *sin|------------|| |      2   *cos|------------|   I*2   *sin|------------||
|            \     2      /             \     2      /| |            \     2      /             \     2      /| |              \     2      /             \     2      /| |              \     2      /             \     2      /|
|x + ---------------------- + ------------------------|*|x + ---------------------- - ------------------------|*|x + - ---------------------- + ------------------------|*|x + - ---------------------- - ------------------------|
\              2                         2            / \              2                         2            / \                2                         2            / \                2                         2            /
$$\left(x + \left(\frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2} - \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(\frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2} + \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(- \frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2} + \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(- \frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2} - \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}}{2}\right)\right)$$ 
(((x + 2^(3/4)*cos(atan(sqrt(31))/2)/2 + i*2^(3/4)*sin(atan(sqrt(31))/2)/2)*(x + 2^(3/4)*cos(atan(sqrt(31))/2)/2 - i*2^(3/4)*sin(atan(sqrt(31))/2)/2))*(x - 2^(3/4)*cos(atan(sqrt(31))/2)/2 + i*2^(3/4)*sin(atan(sqrt(31))/2)/2))*(x - 2^(3/4)*cos(atan(sqrt(31))/2)/2 - i*2^(3/4)*sin(atan(sqrt(31))/2)/2)
General simplification [src] 
     2      4
2 - x  + 4*x
$$4 x^{4} - x^{2} + 2$$ 
2 - x^2 + 4*x^4
Numerical answer [src] 
2.0 - x^2 + 4.0*x^4
2.0 - x^2 + 4.0*x^4
Rational denominator [src] 
     2      4
2 - x  + 4*x
$$4 x^{4} - x^{2} + 2$$ 
2 - x^2 + 4*x^4
Combinatorics [src] 
     2      4
2 - x  + 4*x
$$4 x^{4} - x^{2} + 2$$ 
2 - x^2 + 4*x^4
Combining rational expressions [src] 
     2 /        2\
2 + x *\-1 + 4*x /
$$x^{2} \left(4 x^{2} - 1\right) + 2$$ 
2 + x^2*(-1 + 4*x^2)
Assemble expression [src] 
     2      4
2 - x  + 4*x
$$4 x^{4} - x^{2} + 2$$ 
2 - x^2 + 4*x^4
Common denominator [src] 
     2      4
2 - x  + 4*x
$$4 x^{4} - x^{2} + 2$$ 
2 - x^2 + 4*x^4
Powers [src] 
     2      4
2 - x  + 4*x
$$4 x^{4} - x^{2} + 2$$ 
2 - x^2 + 4*x^4
Trigonometric part [src] 
     2      4
2 - x  + 4*x
$$4 x^{4} - x^{2} + 2$$ 
2 - x^2 + 4*x^4
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