Mister Exam
Factor -y^4+9*y^2-1 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 9 y^{2}\right) - 1$$
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
$$a = -1$$
$$b = 9$$
$$c = -1$$
Then
$$m = - \frac{9}{2}$$
$$n = \frac{77}{4}$$
So,
$$\frac{77}{4} - \left(y^{2} - \frac{9}{2}\right)^{2}$$
$$\left(- y^{4} + 9 y^{2}\right) - 1$$
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
$$a = -1$$
$$b = 9$$
$$c = -1$$
Then
$$m = - \frac{9}{2}$$
$$n = \frac{77}{4}$$
So,
$$\frac{77}{4} - \left(y^{2} - \frac{9}{2}\right)^{2}$$
Factorization
[src]
/ ____________\ / ____________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | | / 9 \/ 77 | | / 9 \/ 77 | | / 9 \/ 77 | | / 9 \/ 77 | |x + / - - ------ |*|x - / - - ------ |*|x + / - + ------ |*|x - / - + ------ | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - \sqrt{\frac{9}{2} - \frac{\sqrt{77}}{2}}\right) \left(x + \sqrt{\frac{9}{2} - \frac{\sqrt{77}}{2}}\right) \left(x + \sqrt{\frac{\sqrt{77}}{2} + \frac{9}{2}}\right) \left(x - \sqrt{\frac{\sqrt{77}}{2} + \frac{9}{2}}\right)$$
(((x + sqrt(9/2 - sqrt(77)/2))*(x - sqrt(9/2 - sqrt(77)/2)))*(x + sqrt(9/2 + sqrt(77)/2)))*(x - sqrt(9/2 + sqrt(77)/2))
Combining rational expressions
[src]
2 / 2\ -1 + y *\9 - y /
$$y^{2} \left(9 - y^{2}\right) - 1$$
-1 + y^2*(9 - y^2)