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2*x^2-x+1=0

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

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Numerical solution:

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   2            
2*x  - x + 1 = 0
(2x2−x)+1=0\left(2 x^{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=2a = 2
b=−1b = -1
c=1c = 1
, then
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (2) * (1) = -7

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=14+7i4x_{1} = \frac{1}{4} + \frac{\sqrt{7} i}{4}
x2=14−7i4x_{2} = \frac{1}{4} - \frac{\sqrt{7} i}{4}
Vieta's Theorem
rewrite the equation
(2x2−x)+1=0\left(2 x^{2} - x\right) + 1 = 0
of
ax2+bx+c=0a x^{2} + b x + c = 0
as reduced quadratic equation
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
x2−x2+12=0x^{2} - \frac{x}{2} + \frac{1}{2} = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−12p = - \frac{1}{2}
q=caq = \frac{c}{a}
q=12q = \frac{1}{2}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=12x_{1} + x_{2} = \frac{1}{2}
x1x2=12x_{1} x_{2} = \frac{1}{2}
The graph
-4.0-3.0-2.0-1.00.01.02.03.04.0020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
             ___
     1   I*\/ 7 
x1 = - - -------
     4      4
x1=14−7i4x_{1} = \frac{1}{4} - \frac{\sqrt{7} i}{4} 
             ___
     1   I*\/ 7 
x2 = - + -------
     4      4
x2=14+7i4x_{2} = \frac{1}{4} + \frac{\sqrt{7} i}{4} 
x2 = 1/4 + sqrt(7)*i/4
Sum and product of roots [src] 
sum
        ___           ___
1   I*\/ 7    1   I*\/ 7 
- - ------- + - + -------
4      4      4      4
(14−7i4)+(14+7i4)\left(\frac{1}{4} - \frac{\sqrt{7} i}{4}\right) + \left(\frac{1}{4} + \frac{\sqrt{7} i}{4}\right) 
=
1/2
12\frac{1}{2} 
product
/        ___\ /        ___\
|1   I*\/ 7 | |1   I*\/ 7 |
|- - -------|*|- + -------|
\4      4   / \4      4   /
(14−7i4)(14+7i4)\left(\frac{1}{4} - \frac{\sqrt{7} i}{4}\right) \left(\frac{1}{4} + \frac{\sqrt{7} i}{4}\right) 
=
1/2
12\frac{1}{2} 
1/2
Numerical answer [src] 
x1 = 0.25 + 0.661437827766148*i
x2 = 0.25 - 0.661437827766148*i
x2 = 0.25 - 0.661437827766148*i
The graph
2*x^2-x+1=0 equation
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