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3*x^2-x-5=0

3*x^2-x-5=0 equation

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Numerical solution:

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The solution

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3*x  - x - 5 = 0
$$\left(3 x^{2} - x\right) - 5 = 0$$
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
$$a = 3$$
$$b = -1$$
$$c = -5$$
, then
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (3) * (-5) = 61

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

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

or
$$x_{1} = \frac{1}{6} + \frac{\sqrt{61}}{6}$$
$$x_{2} = \frac{1}{6} - \frac{\sqrt{61}}{6}$$
Vieta's Theorem
rewrite the equation
$$\left(3 x^{2} - x\right) - 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{3} - \frac{5}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{3}$$
$$x_{1} x_{2} = - \frac{5}{3}$$
The graph
Sum and product of roots [src]
sum
      ____         ____
1   \/ 61    1   \/ 61 
- - ------ + - + ------
6     6      6     6   
$$\left(\frac{1}{6} - \frac{\sqrt{61}}{6}\right) + \left(\frac{1}{6} + \frac{\sqrt{61}}{6}\right)$$
=
1/3
$$\frac{1}{3}$$
product
/      ____\ /      ____\
|1   \/ 61 | |1   \/ 61 |
|- - ------|*|- + ------|
\6     6   / \6     6   /
$$\left(\frac{1}{6} - \frac{\sqrt{61}}{6}\right) \left(\frac{1}{6} + \frac{\sqrt{61}}{6}\right)$$
=
-5/3
$$- \frac{5}{3}$$
-5/3
Rapid solution [src]
           ____
     1   \/ 61 
x1 = - - ------
     6     6   
$$x_{1} = \frac{1}{6} - \frac{\sqrt{61}}{6}$$
           ____
     1   \/ 61 
x2 = - + ------
     6     6   
$$x_{2} = \frac{1}{6} + \frac{\sqrt{61}}{6}$$
x2 = 1/6 + sqrt(61)/6
Numerical answer [src]
x1 = -1.13504161265111
x2 = 1.46837494598444
x2 = 1.46837494598444
The graph
3*x^2-x-5=0 equation