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Expression simplification/ Factor perfect squares/ x^2+x-1

Factor x^2+x-1 squared

An expression to simplify:

The solution

You have entered [src] 
 2        
x  + x - 1
(x2+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
(x2+x)−1\left(x^{2} + x\right) - 1
To do this, let's use the formula
ax2+bx+c=a(m+x)2+na x^{2} + b x + c = a \left(m + x\right)^{2} + n
where
m=b2am = \frac{b}{2 a}
n=4ac−b24an = \frac{4 a c - b^{2}}{4 a}
In this case
a=1a = 1
b=1b = 1
c=−1c = -1
Then
m=12m = \frac{1}{2}
n=−54n = - \frac{5}{4}
So,
(x+12)2−54\left(x + \frac{1}{2}\right)^{2} - \frac{5}{4}
General simplification [src] 
          2
-1 + x + x
x2+x−1x^{2} + x - 1 
-1 + x + x^2
Factorization [src] 
/          ___\ /          ___\
|    1   \/ 5 | |    1   \/ 5 |
|x + - - -----|*|x + - + -----|
\    2     2  / \    2     2  /
(x+(12−52))(x+(12+52))\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)
Trigonometric part [src] 
          2
-1 + x + x
x2+x−1x^{2} + x - 1 
-1 + x + x^2
Assemble expression [src] 
          2
-1 + x + x
x2+x−1x^{2} + x - 1 
-1 + x + x^2
Combinatorics [src] 
          2
-1 + x + x
x2+x−1x^{2} + x - 1 
-1 + x + x^2
Rational denominator [src] 
          2
-1 + x + x
x2+x−1x^{2} + x - 1 
-1 + x + x^2
Common denominator [src] 
          2
-1 + x + x
x2+x−1x^{2} + x - 1 
-1 + x + x^2
Numerical answer [src] 
-1.0 + x + x^2
-1.0 + x + x^2
Combining rational expressions [src] 
-1 + x*(1 + x)
x(x+1)−1x \left(x + 1\right) - 1 
-1 + x*(1 + x)
Powers [src] 
          2
-1 + x + x
x2+x−1x^{2} + x - 1 
-1 + x + x^2
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