Mister Exam
Factor -y^4+6*y^2-4 squared
The solution
Factorization
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/ ___________\ / ___________\ / ___________\ / ___________\ | / ___ | | / ___ | | / ___ | | / ___ | \x + \/ 3 - \/ 5 /*\x - \/ 3 - \/ 5 /*\x + \/ 3 + \/ 5 /*\x - \/ 3 + \/ 5 /
$$\left(x - \sqrt{3 - \sqrt{5}}\right) \left(x + \sqrt{3 - \sqrt{5}}\right) \left(x + \sqrt{\sqrt{5} + 3}\right) \left(x - \sqrt{\sqrt{5} + 3}\right)$$
(((x + sqrt(3 - sqrt(5)))*(x - sqrt(3 - sqrt(5))))*(x + sqrt(3 + sqrt(5))))*(x - sqrt(3 + sqrt(5)))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 6 y^{2}\right) - 4$$
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
$$a = -1$$
$$b = 6$$
$$c = -4$$
Then
$$m = -3$$
$$n = 5$$
So,
$$5 - \left(y^{2} - 3\right)^{2}$$
$$\left(- y^{4} + 6 y^{2}\right) - 4$$
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
$$a = -1$$
$$b = 6$$
$$c = -4$$
Then
$$m = -3$$
$$n = 5$$
So,
$$5 - \left(y^{2} - 3\right)^{2}$$
Combining rational expressions
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2 / 2\ -4 + y *\6 - y /
$$y^{2} \left(6 - y^{2}\right) - 4$$
-4 + y^2*(6 - y^2)