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5x^2+3x+6=0

5x^2+3x+6=0 equation

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Numerical solution:

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The solution

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5*x  + 3*x + 6 = 0
$$5 x^{2} + 3 x + 6 = 0$$
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
$$a = 5$$
$$b = 3$$
$$c = 6$$
, then
D = b^2 - 4 * a * c = 

(3)^2 - 4 * (5) * (6) = -111

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
$$x_{1} = - \frac{3}{10} + \frac{\sqrt{111} i}{10}$$
Simplify
$$x_{2} = - \frac{3}{10} - \frac{\sqrt{111} i}{10}$$
Simplify
Vieta's Theorem
rewrite the equation
$$5 x^{2} + 3 x + 6 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{3 x}{5} + \frac{6}{5} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{3}{5}$$
$$q = \frac{c}{a}$$
$$q = \frac{6}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{3}{5}$$
$$x_{1} x_{2} = \frac{6}{5}$$
The graph
Rapid solution [src]
                _____
       3    I*\/ 111 
x1 = - -- - ---------
       10       10   
$$x_{1} = - \frac{3}{10} - \frac{\sqrt{111} i}{10}$$
                _____
       3    I*\/ 111 
x2 = - -- + ---------
       10       10   
$$x_{2} = - \frac{3}{10} + \frac{\sqrt{111} i}{10}$$
Sum and product of roots [src]
sum
               _____              _____
      3    I*\/ 111      3    I*\/ 111 
0 + - -- - --------- + - -- + ---------
      10       10        10       10   
$$\left(0 - \left(\frac{3}{10} + \frac{\sqrt{111} i}{10}\right)\right) - \left(\frac{3}{10} - \frac{\sqrt{111} i}{10}\right)$$
=
-3/5
$$- \frac{3}{5}$$
product
  /           _____\ /           _____\
  |  3    I*\/ 111 | |  3    I*\/ 111 |
1*|- -- - ---------|*|- -- + ---------|
  \  10       10   / \  10       10   /
$$1 \left(- \frac{3}{10} - \frac{\sqrt{111} i}{10}\right) \left(- \frac{3}{10} + \frac{\sqrt{111} i}{10}\right)$$
=
6/5
$$\frac{6}{5}$$
6/5
Numerical answer [src]
x1 = -0.3 - 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i
The graph
5x^2+3x+6=0 equation