Mister Exam
Factor y^4+9*y^2-5 squared
The solution
Factorization
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/ _____________\ / _____________\ / _______________\ / _______________\ | / _____ | | / _____ | | / _____ | | / _____ | | / 9 \/ 101 | | / 9 \/ 101 | | / 9 \/ 101 | | / 9 \/ 101 | |x + I* / - + ------- |*|x - I* / - + ------- |*|x + / - - + ------- |*|x - / - - + ------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{\frac{9}{2} + \frac{\sqrt{101}}{2}}\right) \left(x + i \sqrt{\frac{9}{2} + \frac{\sqrt{101}}{2}}\right) \left(x + \sqrt{- \frac{9}{2} + \frac{\sqrt{101}}{2}}\right) \left(x - \sqrt{- \frac{9}{2} + \frac{\sqrt{101}}{2}}\right)$$
(((x + i*sqrt(9/2 + sqrt(101)/2))*(x - i*sqrt(9/2 + sqrt(101)/2)))*(x + sqrt(-9/2 + sqrt(101)/2)))*(x - sqrt(-9/2 + sqrt(101)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 9 y^{2}\right) - 5$$
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
$$a = 1$$
$$b = 9$$
$$c = -5$$
Then
$$m = \frac{9}{2}$$
$$n = - \frac{101}{4}$$
So,
$$\left(y^{2} + \frac{9}{2}\right)^{2} - \frac{101}{4}$$
$$\left(y^{4} + 9 y^{2}\right) - 5$$
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
$$a = 1$$
$$b = 9$$
$$c = -5$$
Then
$$m = \frac{9}{2}$$
$$n = - \frac{101}{4}$$
So,
$$\left(y^{2} + \frac{9}{2}\right)^{2} - \frac{101}{4}$$
Combining rational expressions
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2 / 2\ -5 + y *\9 + y /
$$y^{2} \left(y^{2} + 9\right) - 5$$
-5 + y^2*(9 + y^2)