Mister Exam
Factor -y^4+9*y^2+6 squared
The solution
Factorization
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/ _______________\ / _______________\ / _____________\ / _____________\ | / _____ | | / _____ | | / _____ | | / _____ | | / 9 \/ 105 | | / 9 \/ 105 | | / 9 \/ 105 | | / 9 \/ 105 | |x + I* / - - + ------- |*|x - I* / - - + ------- |*|x + / - + ------- |*|x - / - + ------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{- \frac{9}{2} + \frac{\sqrt{105}}{2}}\right) \left(x + i \sqrt{- \frac{9}{2} + \frac{\sqrt{105}}{2}}\right) \left(x + \sqrt{\frac{9}{2} + \frac{\sqrt{105}}{2}}\right) \left(x - \sqrt{\frac{9}{2} + \frac{\sqrt{105}}{2}}\right)$$
(((x + i*sqrt(-9/2 + sqrt(105)/2))*(x - i*sqrt(-9/2 + sqrt(105)/2)))*(x + sqrt(9/2 + sqrt(105)/2)))*(x - sqrt(9/2 + sqrt(105)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 9 y^{2}\right) + 6$$
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
$$a = -1$$
$$b = 9$$
$$c = 6$$
Then
$$m = - \frac{9}{2}$$
$$n = \frac{105}{4}$$
So,
$$\frac{105}{4} - \left(y^{2} - \frac{9}{2}\right)^{2}$$
$$\left(- y^{4} + 9 y^{2}\right) + 6$$
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
$$a = -1$$
$$b = 9$$
$$c = 6$$
Then
$$m = - \frac{9}{2}$$
$$n = \frac{105}{4}$$
So,
$$\frac{105}{4} - \left(y^{2} - \frac{9}{2}\right)^{2}$$
Combining rational expressions
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2 / 2\ 6 + y *\9 - y /
$$y^{2} \left(9 - y^{2}\right) + 6$$
6 + y^2*(9 - y^2)