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Identical expressions
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Similar expressions
- 12x^3-3x=0
12x^3+3x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$12 x^{3} + 3 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(12 x^{2} + 3\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$12 x^{2} + 3 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 12$$
$$b = 0$$
$$c = 3$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = \frac{i}{2}$$
$$x_{3} = - \frac{i}{2}$$
The final answer for 12*x^3 + 3*x = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{i}{2}$$
$$x_{3} = - \frac{i}{2}$$
$$12 x^{3} + 3 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(12 x^{2} + 3\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$12 x^{2} + 3 = 0$$
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:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 12$$
$$b = 0$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (12) * (3) = -144
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = \frac{i}{2}$$
$$x_{3} = - \frac{i}{2}$$
The final answer for 12*x^3 + 3*x = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{i}{2}$$
$$x_{3} = - \frac{i}{2}$$
Vieta's Theorem
rewrite the equation
$$12 x^{3} + 3 x = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{x}{4} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{4}$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{4}$$
$$x_{1} x_{2} x_{3} = 0$$
$$12 x^{3} + 3 x = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{x}{4} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{4}$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{4}$$
$$x_{1} x_{2} x_{3} = 0$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
-I
x2 = ---
2
$$x_{2} = - \frac{i}{2}$$
I
x3 = -
2
$$x_{3} = \frac{i}{2}$$
x3 = i/2
Sum and product of roots
[src]
sum
I I - - + - 2 2
$$- \frac{i}{2} + \frac{i}{2}$$
=
0
$$0$$
product
-I I 0*---*- 2 2
$$\frac{i}{2} \cdot 0 \left(- \frac{i}{2}\right)$$
=
0
$$0$$
0