Mister Exam
Factor y^2+y+7 squared
The solution
Factorization
[src]
/ ___\ / ___\ | 1 3*I*\/ 3 | | 1 3*I*\/ 3 | |x + - + ---------|*|x + - - ---------| \ 2 2 / \ 2 2 /
$$\left(x + \left(\frac{1}{2} - \frac{3 \sqrt{3} i}{2}\right)\right) \left(x + \left(\frac{1}{2} + \frac{3 \sqrt{3} i}{2}\right)\right)$$
(x + 1/2 + 3*i*sqrt(3)/2)*(x + 1/2 - 3*i*sqrt(3)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{2} + y\right) + 7$$
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
$$a = 1$$
$$b = 1$$
$$c = 7$$
Then
$$m = \frac{1}{2}$$
$$n = \frac{27}{4}$$
So,
$$\left(y + \frac{1}{2}\right)^{2} + \frac{27}{4}$$
$$\left(y^{2} + y\right) + 7$$
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
$$a = 1$$
$$b = 1$$
$$c = 7$$
Then
$$m = \frac{1}{2}$$
$$n = \frac{27}{4}$$
So,
$$\left(y + \frac{1}{2}\right)^{2} + \frac{27}{4}$$