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3x^2+2=-5x

3x^2+2=-5x equation

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Numerical solution:

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The solution

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   2           
3*x  + 2 = -5*x
$$3 x^{2} + 2 = - 5 x$$
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$3 x^{2} + 2 = - 5 x$$
to
$$5 x + \left(3 x^{2} + 2\right) = 0$$
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
$$a = 3$$
$$b = 5$$
$$c = 2$$
, then
D = b^2 - 4 * a * c = 

(5)^2 - 4 * (3) * (2) = 1

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

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

or
$$x_{1} = - \frac{2}{3}$$
$$x_{2} = -1$$
Vieta's Theorem
rewrite the equation
$$3 x^{2} + 2 = - 5 x$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{5 x}{3} + \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{5}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{5}{3}$$
$$x_{1} x_{2} = \frac{2}{3}$$
The graph
Rapid solution [src]
x1 = -1
$$x_{1} = -1$$
x2 = -2/3
$$x_{2} = - \frac{2}{3}$$
x2 = -2/3
Sum and product of roots [src]
sum
-1 - 2/3
$$-1 - \frac{2}{3}$$
=
-5/3
$$- \frac{5}{3}$$
product
-(-2) 
------
  3   
$$- \frac{-2}{3}$$
=
2/3
$$\frac{2}{3}$$
2/3
Numerical answer [src]
x1 = -0.666666666666667
x2 = -1.0
x2 = -1.0
The graph
3x^2+2=-5x equation