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Expression simplification/ Factor perfect squares/ y^4-9*y^2-3

Factor y^4-9*y^2-3 squared

An expression to simplify:

The solution

You have entered [src] 
 4      2    
y  - 9*y  - 3
$$\left(y^{4} - 9 y^{2}\right) - 3$$ 
y^4 - 9*y^2 - 3
General simplification [src] 
      4      2
-3 + y  - 9*y
$$y^{4} - 9 y^{2} - 3$$ 
-3 + y^4 - 9*y^2
Factorization [src] 
/           ______________\ /           ______________\ /         ____________\ /         ____________\
|          /         ____ | |          /         ____ | |        /       ____ | |        /       ____ |
|         /    9   \/ 93  | |         /    9   \/ 93  | |       /  9   \/ 93  | |       /  9   \/ 93  |
|x + I*  /   - - + ------ |*|x - I*  /   - - + ------ |*|x +   /   - + ------ |*|x -   /   - + ------ |
\      \/      2     2    / \      \/      2     2    / \    \/    2     2    / \    \/    2     2    /
$$\left(x - i \sqrt{- \frac{9}{2} + \frac{\sqrt{93}}{2}}\right) \left(x + i \sqrt{- \frac{9}{2} + \frac{\sqrt{93}}{2}}\right) \left(x + \sqrt{\frac{9}{2} + \frac{\sqrt{93}}{2}}\right) \left(x - \sqrt{\frac{9}{2} + \frac{\sqrt{93}}{2}}\right)$$ 
(((x + i*sqrt(-9/2 + sqrt(93)/2))*(x - i*sqrt(-9/2 + sqrt(93)/2)))*(x + sqrt(9/2 + sqrt(93)/2)))*(x - sqrt(9/2 + sqrt(93)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 9 y^{2}\right) - 3$$
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
$$a = 1$$
$$b = -9$$
$$c = -3$$
Then
$$m = - \frac{9}{2}$$
$$n = - \frac{93}{4}$$
So,
$$\left(y^{2} - \frac{9}{2}\right)^{2} - \frac{93}{4}$$
Rational denominator [src] 
      4      2
-3 + y  - 9*y
$$y^{4} - 9 y^{2} - 3$$ 
-3 + y^4 - 9*y^2
Trigonometric part [src] 
      4      2
-3 + y  - 9*y
$$y^{4} - 9 y^{2} - 3$$ 
-3 + y^4 - 9*y^2
Powers [src] 
      4      2
-3 + y  - 9*y
$$y^{4} - 9 y^{2} - 3$$ 
-3 + y^4 - 9*y^2
Assemble expression [src] 
      4      2
-3 + y  - 9*y
$$y^{4} - 9 y^{2} - 3$$ 
-3 + y^4 - 9*y^2
Combining rational expressions [src] 
      2 /      2\
-3 + y *\-9 + y /
$$y^{2} \left(y^{2} - 9\right) - 3$$ 
-3 + y^2*(-9 + y^2)
Numerical answer [src] 
-3.0 + y^4 - 9.0*y^2
-3.0 + y^4 - 9.0*y^2
Combinatorics [src] 
      4      2
-3 + y  - 9*y
$$y^{4} - 9 y^{2} - 3$$ 
-3 + y^4 - 9*y^2
Common denominator [src] 
      4      2
-3 + y  - 9*y
$$y^{4} - 9 y^{2} - 3$$ 
-3 + y^4 - 9*y^2
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