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3x^2-4x+5 equation

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Numerical solution:

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The solution

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3*x  - 4*x + 5 = 0
(3x2−4x)+5=0\left(3 x^{2} - 4 x\right) + 5 = 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=3a = 3
b=−4b = -4
c=5c = 5
, then
D = b^2 - 4 * a * c = 

(-4)^2 - 4 * (3) * (5) = -44

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=23+11i3x_{1} = \frac{2}{3} + \frac{\sqrt{11} i}{3}
x2=23−11i3x_{2} = \frac{2}{3} - \frac{\sqrt{11} i}{3}
Vieta's Theorem
rewrite the equation
(3x2−4x)+5=0\left(3 x^{2} - 4 x\right) + 5 = 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
x2−4x3+53=0x^{2} - \frac{4 x}{3} + \frac{5}{3} = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−43p = - \frac{4}{3}
q=caq = \frac{c}{a}
q=53q = \frac{5}{3}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=43x_{1} + x_{2} = \frac{4}{3}
x1x2=53x_{1} x_{2} = \frac{5}{3}
The graph
-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.54.0020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
             ____
     2   I*\/ 11 
x1 = - - --------
     3      3
x1=23−11i3x_{1} = \frac{2}{3} - \frac{\sqrt{11} i}{3} 
             ____
     2   I*\/ 11 
x2 = - + --------
     3      3
x2=23+11i3x_{2} = \frac{2}{3} + \frac{\sqrt{11} i}{3} 
x2 = 2/3 + sqrt(11)*i/3
Sum and product of roots [src] 
sum
        ____           ____
2   I*\/ 11    2   I*\/ 11 
- - -------- + - + --------
3      3       3      3
(23−11i3)+(23+11i3)\left(\frac{2}{3} - \frac{\sqrt{11} i}{3}\right) + \left(\frac{2}{3} + \frac{\sqrt{11} i}{3}\right) 
=
4/3
43\frac{4}{3} 
product
/        ____\ /        ____\
|2   I*\/ 11 | |2   I*\/ 11 |
|- - --------|*|- + --------|
\3      3    / \3      3    /
(23−11i3)(23+11i3)\left(\frac{2}{3} - \frac{\sqrt{11} i}{3}\right) \left(\frac{2}{3} + \frac{\sqrt{11} i}{3}\right) 
=
5/3
53\frac{5}{3} 
5/3
Numerical answer [src] 
x1 = 0.666666666666667 + 1.10554159678513*i
x2 = 0.666666666666667 - 1.10554159678513*i
x2 = 0.666666666666667 - 1.10554159678513*i
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