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Expression simplification/ Factor perfect squares/ t^4+t^2-4

Factor t^4+t^2-4 squared

An expression to simplify:

The solution

You have entered [src] 
 4    2    
t  + t  - 4
$$\left(t^{4} + t^{2}\right) - 4$$ 
t^4 + t^2 - 4
Factorization [src] 
/           ____________\ /           ____________\ /         ______________\ /         ______________\
|          /       ____ | |          /       ____ | |        /         ____ | |        /         ____ |
|         /  1   \/ 17  | |         /  1   \/ 17  | |       /    1   \/ 17  | |       /    1   \/ 17  |
|t + I*  /   - + ------ |*|t - I*  /   - + ------ |*|t +   /   - - + ------ |*|t -   /   - - + ------ |
\      \/    2     2    / \      \/    2     2    / \    \/      2     2    / \    \/      2     2    /
$$\left(t - i \sqrt{\frac{1}{2} + \frac{\sqrt{17}}{2}}\right) \left(t + i \sqrt{\frac{1}{2} + \frac{\sqrt{17}}{2}}\right) \left(t + \sqrt{- \frac{1}{2} + \frac{\sqrt{17}}{2}}\right) \left(t - \sqrt{- \frac{1}{2} + \frac{\sqrt{17}}{2}}\right)$$ 
(((t + i*sqrt(1/2 + sqrt(17)/2))*(t - i*sqrt(1/2 + sqrt(17)/2)))*(t + sqrt(-1/2 + sqrt(17)/2)))*(t - sqrt(-1/2 + sqrt(17)/2))
General simplification [src] 
      2    4
-4 + t  + t
$$t^{4} + t^{2} - 4$$ 
-4 + t^2 + t^4
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(t^{4} + t^{2}\right) - 4$$
To do this, let's use the formula
$$a t^{4} + b t^{2} + c = a \left(m + t^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 1$$
$$c = -4$$
Then
$$m = \frac{1}{2}$$
$$n = - \frac{17}{4}$$
So,
$$\left(t^{2} + \frac{1}{2}\right)^{2} - \frac{17}{4}$$
Common denominator [src] 
      2    4
-4 + t  + t
$$t^{4} + t^{2} - 4$$ 
-4 + t^2 + t^4
Rational denominator [src] 
      2    4
-4 + t  + t
$$t^{4} + t^{2} - 4$$ 
-4 + t^2 + t^4
Assemble expression [src] 
      2    4
-4 + t  + t
$$t^{4} + t^{2} - 4$$ 
-4 + t^2 + t^4
Trigonometric part [src] 
      2    4
-4 + t  + t
$$t^{4} + t^{2} - 4$$ 
-4 + t^2 + t^4
Numerical answer [src] 
-4.0 + t^2 + t^4
-4.0 + t^2 + t^4
Powers [src] 
      2    4
-4 + t  + t
$$t^{4} + t^{2} - 4$$ 
-4 + t^2 + t^4
Combining rational expressions [src] 
      2 /     2\
-4 + t *\1 + t /
$$t^{2} \left(t^{2} + 1\right) - 4$$ 
-4 + t^2*(1 + t^2)
Combinatorics [src] 
      2    4
-4 + t  + t
$$t^{4} + t^{2} - 4$$ 
-4 + t^2 + t^4
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