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x^2+x-5=0

x^2+x-5=0 equation

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Numerical solution:

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The solution

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 2            
x  + x - 5 = 0
$$\left(x^{2} + x\right) - 5 = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = -5$$
, then
D = b^2 - 4 * a * c = 

(1)^2 - 4 * (1) * (-5) = 21

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

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

or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{21}}{2}$$
$$x_{2} = - \frac{\sqrt{21}}{2} - \frac{1}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = -5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = -5$$
The graph
Sum and product of roots [src]
sum
        ____           ____
  1   \/ 21      1   \/ 21 
- - + ------ + - - - ------
  2     2        2     2   
$$\left(- \frac{\sqrt{21}}{2} - \frac{1}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{21}}{2}\right)$$
=
-1
$$-1$$
product
/        ____\ /        ____\
|  1   \/ 21 | |  1   \/ 21 |
|- - + ------|*|- - - ------|
\  2     2   / \  2     2   /
$$\left(- \frac{1}{2} + \frac{\sqrt{21}}{2}\right) \left(- \frac{\sqrt{21}}{2} - \frac{1}{2}\right)$$
=
-5
$$-5$$
-5
Rapid solution [src]
             ____
       1   \/ 21 
x1 = - - + ------
       2     2   
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{21}}{2}$$
             ____
       1   \/ 21 
x2 = - - - ------
       2     2   
$$x_{2} = - \frac{\sqrt{21}}{2} - \frac{1}{2}$$
x2 = -sqrt(21)/2 - 1/2
Numerical answer [src]
x1 = -2.79128784747792
x2 = 1.79128784747792
x2 = 1.79128784747792
The graph
x^2+x-5=0 equation