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

Factor polynomial x^2+x-1

An expression to simplify:

The solution

You have entered [src] 
 2        
x  + x - 1
$$\left(x^{2} + x\right) - 1$$ 
x^2 + x - 1
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{2} + x\right) - 1$$
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
$$a = 1$$
$$b = 1$$
$$c = -1$$
Then
$$m = \frac{1}{2}$$
$$n = - \frac{5}{4}$$
So,
$$\left(x + \frac{1}{2}\right)^{2} - \frac{5}{4}$$
Factorization [src] 
/          ___\ /          ___\
|    1   \/ 5 | |    1   \/ 5 |
|x + - - -----|*|x + - + -----|
\    2     2  / \    2     2  /
$$\left(x + \left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right)\right) \left(x + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)\right)$$ 
(x + 1/2 - sqrt(5)/2)*(x + 1/2 + sqrt(5)/2)
General simplification [src] 
          2
-1 + x + x
$$x^{2} + x - 1$$ 
-1 + x + x^2
Numerical answer [src] 
-1.0 + x + x^2
-1.0 + x + x^2
Rational denominator [src] 
          2
-1 + x + x
$$x^{2} + x - 1$$ 
-1 + x + x^2
Powers [src] 
          2
-1 + x + x
$$x^{2} + x - 1$$ 
-1 + x + x^2
Common denominator [src] 
          2
-1 + x + x
$$x^{2} + x - 1$$ 
-1 + x + x^2
Combining rational expressions [src] 
-1 + x*(1 + x)
$$x \left(x + 1\right) - 1$$ 
-1 + x*(1 + x)
Trigonometric part [src] 
          2
-1 + x + x
$$x^{2} + x - 1$$ 
-1 + x + x^2
Combinatorics [src] 
          2
-1 + x + x
$$x^{2} + x - 1$$ 
-1 + x + x^2
Assemble expression [src] 
          2
-1 + x + x
$$x^{2} + x - 1$$ 
-1 + x + x^2
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