/ ____________\ / ____________\ / ____________\ / ____________\
| / ____ | | / ____ | | / ____ | | / ____ |
\x + \/ 5 - \/ 23 /*\x - \/ 5 - \/ 23 /*\x + \/ 5 + \/ 23 /*\x - \/ 5 + \/ 23 /
$$\left(x - \sqrt{5 - \sqrt{23}}\right) \left(x + \sqrt{5 - \sqrt{23}}\right) \left(x + \sqrt{\sqrt{23} + 5}\right) \left(x - \sqrt{\sqrt{23} + 5}\right)$$
(((x + sqrt(5 - sqrt(23)))*(x - sqrt(5 - sqrt(23))))*(x + sqrt(5 + sqrt(23))))*(x - sqrt(5 + sqrt(23)))
General simplification
[src]
$$- y^{4} + 10 y^{2} - 2$$
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 10 y^{2}\right) - 2$$
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
$$a = -1$$
$$b = 10$$
$$c = -2$$
Then
$$m = -5$$
$$n = 23$$
So,
$$23 - \left(y^{2} - 5\right)^{2}$$
Rational denominator
[src]
$$- y^{4} + 10 y^{2} - 2$$
$$- y^{4} + 10 y^{2} - 2$$
$$- y^{4} + 10 y^{2} - 2$$
$$- y^{4} + 10 y^{2} - 2$$
Assemble expression
[src]
$$- y^{4} + 10 y^{2} - 2$$
Combining rational expressions
[src]
$$y^{2} \left(10 - y^{2}\right) - 2$$
$$- y^{4} + 10 y^{2} - 2$$