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

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

An expression to simplify:

The solution

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