Mister Exam
Factor 3*x^2+5*x+3 squared
The solution
Factorization
[src]
/ ____\ / ____\ | 5 I*\/ 11 | | 5 I*\/ 11 | |x + - + --------|*|x + - - --------| \ 6 6 / \ 6 6 /
$$\left(x + \left(\frac{5}{6} - \frac{\sqrt{11} i}{6}\right)\right) \left(x + \left(\frac{5}{6} + \frac{\sqrt{11} i}{6}\right)\right)$$
(x + 5/6 + i*sqrt(11)/6)*(x + 5/6 - i*sqrt(11)/6)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(3 x^{2} + 5 x\right) + 3$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 3$$
$$b = 5$$
$$c = 3$$
Then
$$m = \frac{5}{6}$$
$$n = \frac{11}{12}$$
So,
$$3 \left(x + \frac{5}{6}\right)^{2} + \frac{11}{12}$$
$$\left(3 x^{2} + 5 x\right) + 3$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 3$$
$$b = 5$$
$$c = 3$$
Then
$$m = \frac{5}{6}$$
$$n = \frac{11}{12}$$
So,
$$3 \left(x + \frac{5}{6}\right)^{2} + \frac{11}{12}$$