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sqrt(x-1)=x

sqrt(x-1)=x equation

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Numerical solution:

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The solution

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  _______    
\/ x - 1  = x
x−1=x\sqrt{x - 1} = x
Simplify
Detail solution
Given the equation
x−1=x\sqrt{x - 1} = x
x−1=x\sqrt{x - 1} = x
We raise the equation sides to 2-th degree
x−1=x2x - 1 = x^{2}
x−1=x2x - 1 = x^{2}
Transfer the right side of the equation left part with negative sign
−x2+x−1=0- x^{2} + x - 1 = 0
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=1b = 1
c=−1c = -1
, then
D = b^2 - 4 * a * c = 

(1)^2 - 4 * (-1) * (-1) = -3

Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)

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

or
x1=12−3i2x_{1} = \frac{1}{2} - \frac{\sqrt{3} i}{2}
x2=12+3i2x_{2} = \frac{1}{2} + \frac{\sqrt{3} i}{2}
The graph
02468-8-6-4-2-1010-2020
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
        ___           ___
1   I*\/ 3    1   I*\/ 3 
- - ------- + - + -------
2      2      2      2
(12−3i2)+(12+3i2)\left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right) 
=
1
11 
product
/        ___\ /        ___\
|1   I*\/ 3 | |1   I*\/ 3 |
|- - -------|*|- + -------|
\2      2   / \2      2   /
(12−3i2)(12+3i2)\left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right) 
=
1
11 
1
Rapid solution [src] 
             ___
     1   I*\/ 3 
x1 = - - -------
     2      2
x1=12−3i2x_{1} = \frac{1}{2} - \frac{\sqrt{3} i}{2} 
             ___
     1   I*\/ 3 
x2 = - + -------
     2      2
x2=12+3i2x_{2} = \frac{1}{2} + \frac{\sqrt{3} i}{2} 
x2 = 1/2 + sqrt(3)*i/2
Numerical answer [src] 
x1 = 0.5 - 0.866025403784439*i
x2 = 0.5 + 0.866025403784439*i
x2 = 0.5 + 0.866025403784439*i
The graph
sqrt(x-1)=x equation
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