Mister Exam
Factor -q^2-q-11 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- q^{2} - q\right) - 11$$
To do this, let's use the formula
$$a q^{2} + b q + c = a \left(m + q\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = -1$$
$$c = -11$$
Then
$$m = \frac{1}{2}$$
$$n = - \frac{43}{4}$$
So,
$$- \left(q + \frac{1}{2}\right)^{2} - \frac{43}{4}$$
$$\left(- q^{2} - q\right) - 11$$
To do this, let's use the formula
$$a q^{2} + b q + c = a \left(m + q\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = -1$$
$$c = -11$$
Then
$$m = \frac{1}{2}$$
$$n = - \frac{43}{4}$$
So,
$$- \left(q + \frac{1}{2}\right)^{2} - \frac{43}{4}$$
Factorization
[src]
/ ____\ / ____\ | 1 I*\/ 43 | | 1 I*\/ 43 | |q + - + --------|*|q + - - --------| \ 2 2 / \ 2 2 /
$$\left(q + \left(\frac{1}{2} - \frac{\sqrt{43} i}{2}\right)\right) \left(q + \left(\frac{1}{2} + \frac{\sqrt{43} i}{2}\right)\right)$$
(q + 1/2 + i*sqrt(43)/2)*(q + 1/2 - i*sqrt(43)/2)
Combining rational expressions
[src]
-11 + q*(-1 - q)
$$q \left(- q - 1\right) - 11$$
-11 + q*(-1 - q)