Mister Exam
Factor polynomial 2*x^2+6*x+9
The solution
Factorization
[src]
/ 3 3*I\ / 3 3*I\ |x + - + ---|*|x + - - ---| \ 2 2 / \ 2 2 /
$$\left(x + \left(\frac{3}{2} - \frac{3 i}{2}\right)\right) \left(x + \left(\frac{3}{2} + \frac{3 i}{2}\right)\right)$$
(x + 3/2 + 3*i/2)*(x + 3/2 - 3*i/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(2 x^{2} + 6 x\right) + 9$$
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
$$a = 2$$
$$b = 6$$
$$c = 9$$
Then
$$m = \frac{3}{2}$$
$$n = \frac{9}{2}$$
So,
$$2 \left(x + \frac{3}{2}\right)^{2} + \frac{9}{2}$$
$$\left(2 x^{2} + 6 x\right) + 9$$
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
$$a = 2$$
$$b = 6$$
$$c = 9$$
Then
$$m = \frac{3}{2}$$
$$n = \frac{9}{2}$$
So,
$$2 \left(x + \frac{3}{2}\right)^{2} + \frac{9}{2}$$