Mister Exam
Factor y^4-9*y^2+15 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 9 y^{2}\right) + 15$$
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
$$a = 1$$
$$b = -9$$
$$c = 15$$
Then
$$m = - \frac{9}{2}$$
$$n = - \frac{21}{4}$$
So,
$$\left(y^{2} - \frac{9}{2}\right)^{2} - \frac{21}{4}$$
$$\left(y^{4} - 9 y^{2}\right) + 15$$
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
$$a = 1$$
$$b = -9$$
$$c = 15$$
Then
$$m = - \frac{9}{2}$$
$$n = - \frac{21}{4}$$
So,
$$\left(y^{2} - \frac{9}{2}\right)^{2} - \frac{21}{4}$$
Factorization
[src]
/ ____________\ / ____________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | | / 9 \/ 21 | | / 9 \/ 21 | | / 9 \/ 21 | | / 9 \/ 21 | |x + / - - ------ |*|x - / - - ------ |*|x + / - + ------ |*|x - / - + ------ | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - \sqrt{\frac{9}{2} - \frac{\sqrt{21}}{2}}\right) \left(x + \sqrt{\frac{9}{2} - \frac{\sqrt{21}}{2}}\right) \left(x + \sqrt{\frac{\sqrt{21}}{2} + \frac{9}{2}}\right) \left(x - \sqrt{\frac{\sqrt{21}}{2} + \frac{9}{2}}\right)$$
(((x + sqrt(9/2 - sqrt(21)/2))*(x - sqrt(9/2 - sqrt(21)/2)))*(x + sqrt(9/2 + sqrt(21)/2)))*(x - sqrt(9/2 + sqrt(21)/2))
Combining rational expressions
[src]
2 / 2\ 15 + y *\-9 + y /
$$y^{2} \left(y^{2} - 9\right) + 15$$
15 + y^2*(-9 + y^2)