Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 10/(x+6)=1
-
Equation -2*(-2-y)-y-2=0
-
Equation 2*x^2+18*x=0
- Equation 2ax=6
- Express {x} in terms of y where:
- -4*x-9*y=12
- -2*x+15*y=9
- -1*x+11*y=3
- -5*x+15*y=6
- Graphing y =:
- -x^3+3*x
- Derivative of:
- -x^3+3*x
- Integral of d{x}:
- -x^3+3*x
-
Identical expressions
- -x^ three + three *x= zero
- minus x cubed plus 3 multiply by x equally 0
- minus x to the power of three plus 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:
$$x \left(3 - x^{2}\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$3 - x^{2} = 0$$
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 two roots.
or
$$x_{2} = - \sqrt{3}$$
$$x_{3} = \sqrt{3}$$
The final answer for -x^3 + 3*x = 0:
$$x_{1} = 0$$
$$x_{2} = - \sqrt{3}$$
$$x_{3} = \sqrt{3}$$
$$- x^{3} + 3 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(3 - x^{2}\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$3 - x^{2} = 0$$
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 two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = - \sqrt{3}$$
$$x_{3} = \sqrt{3}$$
The final answer for -x^3 + 3*x = 0:
$$x_{1} = 0$$
$$x_{2} = - \sqrt{3}$$
$$x_{3} = \sqrt{3}$$
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
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = 0$$
Sum and product of roots
[src]
sum
___ ___ - \/ 3 + \/ 3
$$- \sqrt{3} + \sqrt{3}$$
=
0
$$0$$
product
/ ___\ ___ 0*\-\/ 3 /*\/ 3
$$\sqrt{3} \cdot 0 \left(- \sqrt{3}\right)$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
___ x2 = -\/ 3
$$x_{2} = - \sqrt{3}$$
___ x3 = \/ 3
$$x_{3} = \sqrt{3}$$
x3 = sqrt(3)
The graph