Mister Exam
Factor polynomial 3*x^2-2*x+1
The solution
Factorization
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/ ___\ / ___\ | 1 I*\/ 2 | | 1 I*\/ 2 | |x + - - + -------|*|x + - - - -------| \ 3 3 / \ 3 3 /
$$\left(x + \left(- \frac{1}{3} - \frac{\sqrt{2} i}{3}\right)\right) \left(x + \left(- \frac{1}{3} + \frac{\sqrt{2} i}{3}\right)\right)$$
(x - 1/3 + i*sqrt(2)/3)*(x - 1/3 - i*sqrt(2)/3)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(3 x^{2} - 2 x\right) + 1$$
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
$$a = 3$$
$$b = -2$$
$$c = 1$$
Then
$$m = - \frac{1}{3}$$
$$n = \frac{2}{3}$$
So,
$$3 \left(x - \frac{1}{3}\right)^{2} + \frac{2}{3}$$
$$\left(3 x^{2} - 2 x\right) + 1$$
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
$$a = 3$$
$$b = -2$$
$$c = 1$$
Then
$$m = - \frac{1}{3}$$
$$n = \frac{2}{3}$$
So,
$$3 \left(x - \frac{1}{3}\right)^{2} + \frac{2}{3}$$
Combining rational expressions
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1 + x*(-2 + 3*x)
$$x \left(3 x - 2\right) + 1$$
1 + x*(-2 + 3*x)