Mister Exam
Factor -y^2+3*y-12 squared
The solution
Factorization
[src]
/ ____\ / ____\ | 3 I*\/ 39 | | 3 I*\/ 39 | |x + - - + --------|*|x + - - - --------| \ 2 2 / \ 2 2 /
$$\left(x + \left(- \frac{3}{2} - \frac{\sqrt{39} i}{2}\right)\right) \left(x + \left(- \frac{3}{2} + \frac{\sqrt{39} i}{2}\right)\right)$$
(x - 3/2 + i*sqrt(39)/2)*(x - 3/2 - i*sqrt(39)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{2} + 3 y\right) - 12$$
To do this, let's use the formula
$$a y^{2} + b y + c = a \left(m + y\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = 3$$
$$c = -12$$
Then
$$m = - \frac{3}{2}$$
$$n = - \frac{39}{4}$$
So,
$$- \left(y - \frac{3}{2}\right)^{2} - \frac{39}{4}$$
$$\left(- y^{2} + 3 y\right) - 12$$
To do this, let's use the formula
$$a y^{2} + b y + c = a \left(m + y\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = 3$$
$$c = -12$$
Then
$$m = - \frac{3}{2}$$
$$n = - \frac{39}{4}$$
So,
$$- \left(y - \frac{3}{2}\right)^{2} - \frac{39}{4}$$