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x^2-2*x-1=0

x^2-2*x-1=0 equation

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Numerical solution:

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The solution

You have entered [src] 
 2              
x  - 2*x - 1 = 0
(x2−2x)−1=0\left(x^{2} - 2 x\right) - 1 = 0
Simplify
Detail solution
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=−2b = -2
c=−1c = -1
, then
D = b^2 - 4 * a * c = 

(-2)^2 - 4 * (1) * (-1) = 8

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=1+2x_{1} = 1 + \sqrt{2}
x2=1−2x_{2} = 1 - \sqrt{2}
Vieta's Theorem
it is reduced quadratic equation
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−2p = -2
q=caq = \frac{c}{a}
q=−1q = -1
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=2x_{1} + x_{2} = 2
x1x2=−1x_{1} x_{2} = -1
The graph
05-15-10-51015200-100
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
           ___
x1 = 1 - \/ 2
x1=1−2x_{1} = 1 - \sqrt{2} 
           ___
x2 = 1 + \/ 2
x2=1+2x_{2} = 1 + \sqrt{2} 
x2 = 1 + sqrt(2)
Sum and product of roots [src] 
sum
      ___         ___
1 - \/ 2  + 1 + \/ 2
(1−2)+(1+2)\left(1 - \sqrt{2}\right) + \left(1 + \sqrt{2}\right) 
=
2
22 
product
/      ___\ /      ___\
\1 - \/ 2 /*\1 + \/ 2 /
(1−2)(1+2)\left(1 - \sqrt{2}\right) \left(1 + \sqrt{2}\right) 
=
-1
−1-1 
-1
Numerical answer [src] 
x1 = -0.414213562373095
x2 = 2.41421356237309
x2 = 2.41421356237309
The graph
x^2-2*x-1=0 equation
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