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