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x^2+5*x+12=0

x^2+5*x+12=0 equation

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Numerical solution:

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The solution

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 2               
x  + 5*x + 12 = 0
(x2+5x)+12=0\left(x^{2} + 5 x\right) + 12 = 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=5b = 5
c=12c = 12
, then
D = b^2 - 4 * a * c = 

(5)^2 - 4 * (1) * (12) = -23

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=−52+23i2x_{1} = - \frac{5}{2} + \frac{\sqrt{23} i}{2}
x2=−52−23i2x_{2} = - \frac{5}{2} - \frac{\sqrt{23} 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=5p = 5
q=caq = \frac{c}{a}
q=12q = 12
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−5x_{1} + x_{2} = -5
x1x2=12x_{1} x_{2} = 12
The graph
-8.0-7.0-6.0-5.0-4.0-3.0-2.0-1.00.01.0020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
               ____
       5   I*\/ 23 
x1 = - - - --------
       2      2
x1=−52−23i2x_{1} = - \frac{5}{2} - \frac{\sqrt{23} i}{2} 
               ____
       5   I*\/ 23 
x2 = - - + --------
       2      2
x2=−52+23i2x_{2} = - \frac{5}{2} + \frac{\sqrt{23} i}{2} 
x2 = -5/2 + sqrt(23)*i/2
Sum and product of roots [src] 
sum
          ____             ____
  5   I*\/ 23      5   I*\/ 23 
- - - -------- + - - + --------
  2      2         2      2
(−52−23i2)+(−52+23i2)\left(- \frac{5}{2} - \frac{\sqrt{23} i}{2}\right) + \left(- \frac{5}{2} + \frac{\sqrt{23} i}{2}\right) 
=
-5
−5-5 
product
/          ____\ /          ____\
|  5   I*\/ 23 | |  5   I*\/ 23 |
|- - - --------|*|- - + --------|
\  2      2    / \  2      2    /
(−52−23i2)(−52+23i2)\left(- \frac{5}{2} - \frac{\sqrt{23} i}{2}\right) \left(- \frac{5}{2} + \frac{\sqrt{23} i}{2}\right) 
=
12
1212 
12
Numerical answer [src] 
x1 = -2.5 + 2.39791576165636*i
x2 = -2.5 - 2.39791576165636*i
x2 = -2.5 - 2.39791576165636*i
The graph
x^2+5*x+12=0 equation
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