Mister Exam
Factor y^4-6*y^2-5 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 6 y^{2}\right) - 5$$
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
$$a = 1$$
$$b = -6$$
$$c = -5$$
Then
$$m = -3$$
$$n = -14$$
So,
$$\left(y^{2} - 3\right)^{2} - 14$$
$$\left(y^{4} - 6 y^{2}\right) - 5$$
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
$$a = 1$$
$$b = -6$$
$$c = -5$$
Then
$$m = -3$$
$$n = -14$$
So,
$$\left(y^{2} - 3\right)^{2} - 14$$
Factorization
[src]
/ _____________\ / _____________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | \x + I*\/ -3 + \/ 14 /*\x - I*\/ -3 + \/ 14 /*\x + \/ 3 + \/ 14 /*\x - \/ 3 + \/ 14 /
$$\left(x - i \sqrt{-3 + \sqrt{14}}\right) \left(x + i \sqrt{-3 + \sqrt{14}}\right) \left(x + \sqrt{3 + \sqrt{14}}\right) \left(x - \sqrt{3 + \sqrt{14}}\right)$$
(((x + i*sqrt(-3 + sqrt(14)))*(x - i*sqrt(-3 + sqrt(14))))*(x + sqrt(3 + sqrt(14))))*(x - sqrt(3 + sqrt(14)))
Combining rational expressions
[src]
2 / 2\ -5 + y *\-6 + y /
$$y^{2} \left(y^{2} - 6\right) - 5$$
-5 + y^2*(-6 + y^2)