Mister Exam
Factor 2*x^2+3*x+5 squared
The solution
Factorization
[src]
/ ____\ / ____\ | 3 I*\/ 31 | | 3 I*\/ 31 | |x + - + --------|*|x + - - --------| \ 4 4 / \ 4 4 /
$$\left(x + \left(\frac{3}{4} - \frac{\sqrt{31} i}{4}\right)\right) \left(x + \left(\frac{3}{4} + \frac{\sqrt{31} i}{4}\right)\right)$$
(x + 3/4 + i*sqrt(31)/4)*(x + 3/4 - i*sqrt(31)/4)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(2 x^{2} + 3 x\right) + 5$$
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
$$a = 2$$
$$b = 3$$
$$c = 5$$
Then
$$m = \frac{3}{4}$$
$$n = \frac{31}{8}$$
So,
$$2 \left(x + \frac{3}{4}\right)^{2} + \frac{31}{8}$$
$$\left(2 x^{2} + 3 x\right) + 5$$
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
$$a = 2$$
$$b = 3$$
$$c = 5$$
Then
$$m = \frac{3}{4}$$
$$n = \frac{31}{8}$$
So,
$$2 \left(x + \frac{3}{4}\right)^{2} + \frac{31}{8}$$