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Expression simplification/ Factorization polynomials/ x^2+x+1/4

Factor polynomial x^2+x+1/4

An expression to simplify:

The solution

You have entered [src] 
 2       1
x  + x + -
         4
$$\left(x^{2} + x\right) + \frac{1}{4}$$ 
x^2 + x + 1/4
Factorization [src] 
x + 1/2
$$x + \frac{1}{2}$$ 
x + 1/2
General simplification [src] 
1        2
- + x + x 
4
$$x^{2} + x + \frac{1}{4}$$ 
1/4 + x + x^2
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{2} + x\right) + \frac{1}{4}$$
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
$$a = 1$$
$$b = 1$$
$$c = \frac{1}{4}$$
Then
$$m = \frac{1}{2}$$
$$n = 0$$
So,
$$\left(x + \frac{1}{2}\right)^{2}$$
Assemble expression [src] 
1        2
- + x + x 
4
$$x^{2} + x + \frac{1}{4}$$ 
1/4 + x + x^2
Numerical answer [src] 
0.25 + x + x^2
0.25 + x + x^2
Combinatorics [src] 
         2
(1 + 2*x) 
----------
    4
$$\frac{\left(2 x + 1\right)^{2}}{4}$$ 
(1 + 2*x)^2/4
Combining rational expressions [src] 
1 + 4*x*(1 + x)
---------------
       4
$$\frac{4 x \left(x + 1\right) + 1}{4}$$ 
(1 + 4*x*(1 + x))/4
Powers [src] 
1        2
- + x + x 
4
$$x^{2} + x + \frac{1}{4}$$ 
1/4 + x + x^2
Common denominator [src] 
1        2
- + x + x 
4
$$x^{2} + x + \frac{1}{4}$$ 
1/4 + x + x^2
Rational denominator [src] 
             2
1 + 4*x + 4*x 
--------------
      4
$$\frac{4 x^{2} + 4 x + 1}{4}$$ 
(1 + 4*x + 4*x^2)/4
Trigonometric part [src] 
1        2
- + x + x 
4
$$x^{2} + x + \frac{1}{4}$$ 
1/4 + x + x^2
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