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x^2+3*x+18=0

x^2+3*x+18=0 equation

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Numerical solution:

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The solution

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 2               
x  + 3*x + 18 = 0
(x2+3x)+18=0\left(x^{2} + 3 x\right) + 18 = 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=3b = 3
c=18c = 18
, then
D = b^2 - 4 * a * c = 

(3)^2 - 4 * (1) * (18) = -63

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=−32+37i2x_{1} = - \frac{3}{2} + \frac{3 \sqrt{7} i}{2}
x2=−32−37i2x_{2} = - \frac{3}{2} - \frac{3 \sqrt{7} i}{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=3p = 3
q=caq = \frac{c}{a}
q=18q = 18
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−3x_{1} + x_{2} = -3
x1x2=18x_{1} x_{2} = 18
The graph
-4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
                 ___
       3   3*I*\/ 7 
x1 = - - - ---------
       2       2
x1=−32−37i2x_{1} = - \frac{3}{2} - \frac{3 \sqrt{7} i}{2} 
                 ___
       3   3*I*\/ 7 
x2 = - - + ---------
       2       2
x2=−32+37i2x_{2} = - \frac{3}{2} + \frac{3 \sqrt{7} i}{2} 
x2 = -3/2 + 3*sqrt(7)*i/2
Sum and product of roots [src] 
sum
            ___               ___
  3   3*I*\/ 7      3   3*I*\/ 7 
- - - --------- + - - + ---------
  2       2         2       2
(−32−37i2)+(−32+37i2)\left(- \frac{3}{2} - \frac{3 \sqrt{7} i}{2}\right) + \left(- \frac{3}{2} + \frac{3 \sqrt{7} i}{2}\right) 
=
-3
−3-3 
product
/            ___\ /            ___\
|  3   3*I*\/ 7 | |  3   3*I*\/ 7 |
|- - - ---------|*|- - + ---------|
\  2       2    / \  2       2    /
(−32−37i2)(−32+37i2)\left(- \frac{3}{2} - \frac{3 \sqrt{7} i}{2}\right) \left(- \frac{3}{2} + \frac{3 \sqrt{7} i}{2}\right) 
=
18
1818 
18
Numerical answer [src] 
x1 = -1.5 + 3.96862696659689*i
x2 = -1.5 - 3.96862696659689*i
x2 = -1.5 - 3.96862696659689*i
The graph
x^2+3*x+18=0 equation
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