Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 3*x^2=18*x
-
Equation 0*x=5
-
Equation 5,23+x=-7,24
-
Equation 12/x*10=13
- Express {x} in terms of y where:
- 10*x-18*y=-6
- 3*x-12*y=-14
- 16*x+11*y=-14
- -4*x+20*y=-14
-
Identical expressions
- -x^ three - three *x= zero
- minus x cubed minus 3 multiply by x equally 0
- minus x to the power of three minus three multiply by x equally zero
- -x3-3*x=0
- -x³-3*x=0
- -x to the power of 3-3*x=0
- -x^3-3x=0
- -x3-3x=0
- -x^3-3*x=O
-
Similar expressions
- -x^3+3*x=0
- x^3-3*x=0
-x^3-3*x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$- x^{3} - 3 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(- x^{2} - 3\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$- x^{2} - 3 = 0$$
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
$$a = -1$$
$$b = 0$$
$$c = -3$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = - \sqrt{3} i$$
$$x_{3} = \sqrt{3} i$$
The final answer for -x^3 - 3*x = 0:
$$x_{1} = 0$$
$$x_{2} = - \sqrt{3} i$$
$$x_{3} = \sqrt{3} i$$
$$- x^{3} - 3 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(- x^{2} - 3\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$- x^{2} - 3 = 0$$
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
$$a = -1$$
$$b = 0$$
$$c = -3$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-1) * (-3) = -12
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
$$x_{2} = - \sqrt{3} i$$
$$x_{3} = \sqrt{3} i$$
The final answer for -x^3 - 3*x = 0:
$$x_{1} = 0$$
$$x_{2} = - \sqrt{3} i$$
$$x_{3} = \sqrt{3} i$$
Vieta's Theorem
rewrite the equation
$$- x^{3} - 3 x = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 3 x = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = 0$$
$$- x^{3} - 3 x = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 3 x = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = 0$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
___ x2 = -I*\/ 3
$$x_{2} = - \sqrt{3} i$$
___ x3 = I*\/ 3
$$x_{3} = \sqrt{3} i$$
x3 = sqrt(3)*i
Sum and product of roots
[src]
sum
___ ___ - I*\/ 3 + I*\/ 3
$$- \sqrt{3} i + \sqrt{3} i$$
=
0
$$0$$
product
/ ___\ ___ 0*\-I*\/ 3 /*I*\/ 3
$$\sqrt{3} i 0 \left(- \sqrt{3} i\right)$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 1.73205080756888*i
x2 = 0.0
x3 = -1.73205080756888*i
x3 = -1.73205080756888*i