Mister Exam
Factor -y^4-15*y^2+4 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} - 15 y^{2}\right) + 4$$
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
$$a = -1$$
$$b = -15$$
$$c = 4$$
Then
$$m = \frac{15}{2}$$
$$n = \frac{241}{4}$$
So,
$$\frac{241}{4} - \left(y^{2} + \frac{15}{2}\right)^{2}$$
$$\left(- y^{4} - 15 y^{2}\right) + 4$$
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
$$a = -1$$
$$b = -15$$
$$c = 4$$
Then
$$m = \frac{15}{2}$$
$$n = \frac{241}{4}$$
So,
$$\frac{241}{4} - \left(y^{2} + \frac{15}{2}\right)^{2}$$
Factorization
[src]
/ ______________\ / ______________\ / ________________\ / ________________\ | / _____ | | / _____ | | / _____ | | / _____ | | / 15 \/ 241 | | / 15 \/ 241 | | / 15 \/ 241 | | / 15 \/ 241 | |x + I* / -- + ------- |*|x - I* / -- + ------- |*|x + / - -- + ------- |*|x - / - -- + ------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{\frac{15}{2} + \frac{\sqrt{241}}{2}}\right) \left(x + i \sqrt{\frac{15}{2} + \frac{\sqrt{241}}{2}}\right) \left(x + \sqrt{- \frac{15}{2} + \frac{\sqrt{241}}{2}}\right) \left(x - \sqrt{- \frac{15}{2} + \frac{\sqrt{241}}{2}}\right)$$
(((x + i*sqrt(15/2 + sqrt(241)/2))*(x - i*sqrt(15/2 + sqrt(241)/2)))*(x + sqrt(-15/2 + sqrt(241)/2)))*(x - sqrt(-15/2 + sqrt(241)/2))
Combining rational expressions
[src]
2 / 2\ 4 + y *\-15 - y /
$$y^{2} \left(- y^{2} - 15\right) + 4$$
4 + y^2*(-15 - y^2)