Mister Exam
Factor y^4-12*y^2+7 squared
The solution
Factorization
[src]
/ ____________\ / ____________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | \x + \/ 6 - \/ 29 /*\x - \/ 6 - \/ 29 /*\x + \/ 6 + \/ 29 /*\x - \/ 6 + \/ 29 /
$$\left(x - \sqrt{6 - \sqrt{29}}\right) \left(x + \sqrt{6 - \sqrt{29}}\right) \left(x + \sqrt{\sqrt{29} + 6}\right) \left(x - \sqrt{\sqrt{29} + 6}\right)$$
(((x + sqrt(6 - sqrt(29)))*(x - sqrt(6 - sqrt(29))))*(x + sqrt(6 + sqrt(29))))*(x - sqrt(6 + sqrt(29)))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 12 y^{2}\right) + 7$$
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
$$a = 1$$
$$b = -12$$
$$c = 7$$
Then
$$m = -6$$
$$n = -29$$
So,
$$\left(y^{2} - 6\right)^{2} - 29$$
$$\left(y^{4} - 12 y^{2}\right) + 7$$
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
$$a = 1$$
$$b = -12$$
$$c = 7$$
Then
$$m = -6$$
$$n = -29$$
So,
$$\left(y^{2} - 6\right)^{2} - 29$$
Combining rational expressions
[src]
2 / 2\ 7 + y *\-12 + y /
$$y^{2} \left(y^{2} - 12\right) + 7$$
7 + y^2*(-12 + y^2)