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Expression simplification/ Factorization polynomials/ x^2-x+3

Factor polynomial x^2-x+3

An expression to simplify:

The solution

You have entered [src] 
 2        
x  - x + 3
$$\left(x^{2} - x\right) + 3$$ 
x^2 - x + 3
Factorization [src] 
/              ____\ /              ____\
|      1   I*\/ 11 | |      1   I*\/ 11 |
|x + - - + --------|*|x + - - - --------|
\      2      2    / \      2      2    /
$$\left(x + \left(- \frac{1}{2} - \frac{\sqrt{11} i}{2}\right)\right) \left(x + \left(- \frac{1}{2} + \frac{\sqrt{11} i}{2}\right)\right)$$ 
(x - 1/2 + i*sqrt(11)/2)*(x - 1/2 - i*sqrt(11)/2)
General simplification [src] 
     2    
3 + x  - x
$$x^{2} - x + 3$$ 
3 + x^2 - x
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{2} - x\right) + 3$$
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
$$m = - \frac{1}{2}$$
$$n = \frac{11}{4}$$
So,
$$\left(x - \frac{1}{2}\right)^{2} + \frac{11}{4}$$
Assemble expression [src] 
     2    
3 + x  - x
$$x^{2} - x + 3$$ 
3 + x^2 - x
Combinatorics [src] 
     2    
3 + x  - x
$$x^{2} - x + 3$$ 
3 + x^2 - x
Rational denominator [src] 
     2    
3 + x  - x
$$x^{2} - x + 3$$ 
3 + x^2 - x
Numerical answer [src] 
3.0 + x^2 - x
3.0 + x^2 - x
Trigonometric part [src] 
     2    
3 + x  - x
$$x^{2} - x + 3$$ 
3 + x^2 - x
Common denominator [src] 
     2    
3 + x  - x
$$x^{2} - x + 3$$ 
3 + x^2 - x
Combining rational expressions [src] 
3 + x*(-1 + x)
$$x \left(x - 1\right) + 3$$ 
3 + x*(-1 + x)
Powers [src] 
     2    
3 + x  - x
$$x^{2} - x + 3$$ 
3 + x^2 - x
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