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x^2+5x-12=0

x^2+5x-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) = 73

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

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

or
x1=−52+732x_{1} = - \frac{5}{2} + \frac{\sqrt{73}}{2}
x2=−732−52x_{2} = - \frac{\sqrt{73}}{2} - \frac{5}{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
05-25-20-15-10-51015-200200
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
             ____
       5   \/ 73 
x1 = - - + ------
       2     2
x1=−52+732x_{1} = - \frac{5}{2} + \frac{\sqrt{73}}{2} 
             ____
       5   \/ 73 
x2 = - - - ------
       2     2
x2=−732−52x_{2} = - \frac{\sqrt{73}}{2} - \frac{5}{2} 
x2 = -sqrt(73)/2 - 5/2
Sum and product of roots [src] 
sum
        ____           ____
  5   \/ 73      5   \/ 73 
- - + ------ + - - - ------
  2     2        2     2
(−732−52)+(−52+732)\left(- \frac{\sqrt{73}}{2} - \frac{5}{2}\right) + \left(- \frac{5}{2} + \frac{\sqrt{73}}{2}\right) 
=
-5
−5-5 
product
/        ____\ /        ____\
|  5   \/ 73 | |  5   \/ 73 |
|- - + ------|*|- - - ------|
\  2     2   / \  2     2   /
(−52+732)(−732−52)\left(- \frac{5}{2} + \frac{\sqrt{73}}{2}\right) \left(- \frac{\sqrt{73}}{2} - \frac{5}{2}\right) 
=
-12
−12-12 
-12
Numerical answer [src] 
x1 = 1.77200187265877
x2 = -6.77200187265877
x2 = -6.77200187265877
The graph
x^2+5x-12=0 equation
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