Mister Exam
Factor b^2-b+3 squared
The solution
Factorization
[src]
/ ____\ / ____\ | 1 I*\/ 11 | | 1 I*\/ 11 | |b + - - + --------|*|b + - - - --------| \ 2 2 / \ 2 2 /
$$\left(b + \left(- \frac{1}{2} - \frac{\sqrt{11} i}{2}\right)\right) \left(b + \left(- \frac{1}{2} + \frac{\sqrt{11} i}{2}\right)\right)$$
(b - 1/2 + i*sqrt(11)/2)*(b - 1/2 - i*sqrt(11)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(b^{2} - b\right) + 3$$
To do this, let's use the formula
$$a b^{2} + b^{2} + c = a \left(b + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -1$$
$$c = 3$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{11}{4}$$
So,
$$\left(b - \frac{1}{2}\right)^{2} + \frac{11}{4}$$
$$\left(b^{2} - b\right) + 3$$
To do this, let's use the formula
$$a b^{2} + b^{2} + c = a \left(b + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -1$$
$$c = 3$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{11}{4}$$
So,
$$\left(b - \frac{1}{2}\right)^{2} + \frac{11}{4}$$