Mister Exam
Factor polynomial z^2+6*z+1
The solution
Factorization
[src]
/ ___\ / ___\ \x + 3 + 2*\/ 2 /*\x + 3 - 2*\/ 2 /
$$\left(x + \left(3 - 2 \sqrt{2}\right)\right) \left(x + \left(2 \sqrt{2} + 3\right)\right)$$
(x + 3 + 2*sqrt(2))*(x + 3 - 2*sqrt(2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(z^{2} + 6 z\right) + 1$$
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
$$a = 1$$
$$b = 6$$
$$c = 1$$
Then
$$m = 3$$
$$n = -8$$
So,
$$\left(z + 3\right)^{2} - 8$$
$$\left(z^{2} + 6 z\right) + 1$$
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
$$a = 1$$
$$b = 6$$
$$c = 1$$
Then
$$m = 3$$
$$n = -8$$
So,
$$\left(z + 3\right)^{2} - 8$$