Mister Exam
Factor polynomial x+3-x^2
The solution
Factorization
[src]
/ ____\ / ____\ | 1 \/ 13 | | 1 \/ 13 | |x + - - + ------|*|x + - - - ------| \ 2 2 / \ 2 2 /
$$\left(x + \left(- \frac{1}{2} + \frac{\sqrt{13}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{13}}{2} - \frac{1}{2}\right)\right)$$
(x - 1/2 + sqrt(13)/2)*(x - 1/2 - sqrt(13)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$- x^{2} + \left(x + 3\right)$$
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
$$a = -1$$
$$b = 1$$
$$c = 3$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{13}{4}$$
So,
$$\frac{13}{4} - \left(x - \frac{1}{2}\right)^{2}$$
$$- x^{2} + \left(x + 3\right)$$
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
$$a = -1$$
$$b = 1$$
$$c = 3$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{13}{4}$$
So,
$$\frac{13}{4} - \left(x - \frac{1}{2}\right)^{2}$$