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3x^2-4x+1

3x^2-4x+1 equation

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Numerical solution:

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The solution

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3*x  - 4*x + 1 = 0
3x2−4x+1=03 x^{2} - 4 x + 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=3a = 3
b=−4b = -4
c=1c = 1
, then
D = b^2 - 4 * a * c = 

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

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

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

or
x1=1x_{1} = 1
Simplify
x2=13x_{2} = \frac{1}{3}
Simplify
Vieta's Theorem
rewrite the equation
3x2−4x+1=03 x^{2} - 4 x + 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
x2−4x3+13=0x^{2} - \frac{4 x}{3} + \frac{1}{3} = 0
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=−43p = - \frac{4}{3}
q=caq = \frac{c}{a}
q=13q = \frac{1}{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=13x_{1} x_{2} = \frac{1}{3}
The graph
-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.515.0-500500
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = 1/3
x1=13x_{1} = \frac{1}{3}
Simplify 
x2 = 1
x2=1x_{2} = 1
Simplify
Sum and product of roots [src] 
sum
0 + 1/3 + 1
(0+13)+1\left(0 + \frac{1}{3}\right) + 1 
=
4/3
43\frac{4}{3} 
product
1*1/3*1
1⋅13⋅11 \cdot \frac{1}{3} \cdot 1 
=
1/3
13\frac{1}{3} 
1/3
Numerical answer [src] 
x1 = 0.333333333333333
x2 = 1.0
x2 = 1.0
The graph
3x^2-4x+1 equation
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