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x^2-7*x+3=0

x^2-7*x+3=0 equation

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Numerical solution:

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The solution

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 2              
x  - 7*x + 3 = 0
(x2−7x)+3=0\left(x^{2} - 7 x\right) + 3 = 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=−7b = -7
c=3c = 3
, then
D = b^2 - 4 * a * c = 

(-7)^2 - 4 * (1) * (3) = 37

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

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

or
x1=372+72x_{1} = \frac{\sqrt{37}}{2} + \frac{7}{2}
x2=72−372x_{2} = \frac{7}{2} - \frac{\sqrt{37}}{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=−7p = -7
q=caq = \frac{c}{a}
q=3q = 3
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=7x_{1} + x_{2} = 7
x1x2=3x_{1} x_{2} = 3
The graph
05-10-5101520-200200
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
           ____
     7   \/ 37 
x1 = - - ------
     2     2
x1=72−372x_{1} = \frac{7}{2} - \frac{\sqrt{37}}{2} 
           ____
     7   \/ 37 
x2 = - + ------
     2     2
x2=372+72x_{2} = \frac{\sqrt{37}}{2} + \frac{7}{2} 
x2 = sqrt(37)/2 + 7/2
Sum and product of roots [src] 
sum
      ____         ____
7   \/ 37    7   \/ 37 
- - ------ + - + ------
2     2      2     2
(72−372)+(372+72)\left(\frac{7}{2} - \frac{\sqrt{37}}{2}\right) + \left(\frac{\sqrt{37}}{2} + \frac{7}{2}\right) 
=
7
77 
product
/      ____\ /      ____\
|7   \/ 37 | |7   \/ 37 |
|- - ------|*|- + ------|
\2     2   / \2     2   /
(72−372)(372+72)\left(\frac{7}{2} - \frac{\sqrt{37}}{2}\right) \left(\frac{\sqrt{37}}{2} + \frac{7}{2}\right) 
=
3
33 
3
Numerical answer [src] 
x1 = 0.45861873485089
x2 = 6.54138126514911
x2 = 6.54138126514911
The graph
x^2-7*x+3=0 equation
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