Mister Exam
Factor 2*x^2-x+2 squared
The solution
Factorization
[src]
/ ____\ / ____\ | 1 I*\/ 15 | | 1 I*\/ 15 | |x + - - + --------|*|x + - - - --------| \ 4 4 / \ 4 4 /
$$\left(x + \left(- \frac{1}{4} - \frac{\sqrt{15} i}{4}\right)\right) \left(x + \left(- \frac{1}{4} + \frac{\sqrt{15} i}{4}\right)\right)$$
(x - 1/4 + i*sqrt(15)/4)*(x - 1/4 - i*sqrt(15)/4)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(2 x^{2} - x\right) + 2$$
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
$$a = 2$$
$$b = -1$$
$$c = 2$$
Then
$$m = - \frac{1}{4}$$
$$n = \frac{15}{8}$$
So,
$$2 \left(x - \frac{1}{4}\right)^{2} + \frac{15}{8}$$
$$\left(2 x^{2} - x\right) + 2$$
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
$$a = 2$$
$$b = -1$$
$$c = 2$$
Then
$$m = - \frac{1}{4}$$
$$n = \frac{15}{8}$$
So,
$$2 \left(x - \frac{1}{4}\right)^{2} + \frac{15}{8}$$
Combining rational expressions
[src]
2 + x*(-1 + 2*x)
$$x \left(2 x - 1\right) + 2$$
2 + x*(-1 + 2*x)