Mister Exam
Factor -y^4-9*y^2-4 squared
The solution
Factorization
[src]
/ ____________\ / ____________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | | / 9 \/ 65 | | / 9 \/ 65 | | / 9 \/ 65 | | / 9 \/ 65 | |x + I* / - - ------ |*|x - I* / - - ------ |*|x + I* / - + ------ |*|x - I* / - + ------ | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{\frac{9}{2} - \frac{\sqrt{65}}{2}}\right) \left(x + i \sqrt{\frac{9}{2} - \frac{\sqrt{65}}{2}}\right) \left(x + i \sqrt{\frac{\sqrt{65}}{2} + \frac{9}{2}}\right) \left(x - i \sqrt{\frac{\sqrt{65}}{2} + \frac{9}{2}}\right)$$
(((x + i*sqrt(9/2 - sqrt(65)/2))*(x - i*sqrt(9/2 - sqrt(65)/2)))*(x + i*sqrt(9/2 + sqrt(65)/2)))*(x - i*sqrt(9/2 + sqrt(65)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} - 9 y^{2}\right) - 4$$
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
$$a = -1$$
$$b = -9$$
$$c = -4$$
Then
$$m = \frac{9}{2}$$
$$n = \frac{65}{4}$$
So,
$$\frac{65}{4} - \left(y^{2} + \frac{9}{2}\right)^{2}$$
$$\left(- y^{4} - 9 y^{2}\right) - 4$$
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
$$a = -1$$
$$b = -9$$
$$c = -4$$
Then
$$m = \frac{9}{2}$$
$$n = \frac{65}{4}$$
So,
$$\frac{65}{4} - \left(y^{2} + \frac{9}{2}\right)^{2}$$
Combining rational expressions
[src]
2 / 2\ -4 + y *\-9 - y /
$$y^{2} \left(- y^{2} - 9\right) - 4$$
-4 + y^2*(-9 - y^2)