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x^2+4*x+1=0

x^2+4*x+1=0 equation

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Numerical solution:

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The solution

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

(4)^2 - 4 * (1) * (1) = 12

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

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

or
x1=−2+3x_{1} = -2 + \sqrt{3}
x2=−2−3x_{2} = -2 - \sqrt{3}
Vieta's Theorem
it is reduced quadratic equation
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=4p = 4
q=caq = \frac{c}{a}
q=1q = 1
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−4x_{1} + x_{2} = -4
x1x2=1x_{1} x_{2} = 1
The graph
05-20-15-10-510-200200
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
            ___
x1 = -2 - \/ 3
x1=−2−3x_{1} = -2 - \sqrt{3} 
            ___
x2 = -2 + \/ 3
x2=−2+3x_{2} = -2 + \sqrt{3} 
x2 = -2 + sqrt(3)
Sum and product of roots [src] 
sum
       ___          ___
-2 - \/ 3  + -2 + \/ 3
(−2−3)+(−2+3)\left(-2 - \sqrt{3}\right) + \left(-2 + \sqrt{3}\right) 
=
-4
−4-4 
product
/       ___\ /       ___\
\-2 - \/ 3 /*\-2 + \/ 3 /
(−2−3)(−2+3)\left(-2 - \sqrt{3}\right) \left(-2 + \sqrt{3}\right) 
=
1
11 
1
Numerical answer [src] 
x1 = -0.267949192431123
x2 = -3.73205080756888
x2 = -3.73205080756888
The graph
x^2+4*x+1=0 equation
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