Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation sin(x)=1/5
-
Equation x^2+11=0
-
Equation x^2+5x+6=0
-
Equation -x/7+x=15/7
- Express {x} in terms of y where:
- 9*x+17*y=13
- 20*x+18*y=-9
- -4*x-8*y=-20
- -1*x+4*y=-10
- Graphing y =:
- x^3-12*x
- Limit of the function:
-
x^3-12*x
-
Identical expressions
- x^ three - twelve *x= zero
- x cubed minus 12 multiply by x equally 0
- x to the power of three minus twelve multiply by x equally zero
- x3-12*x=0
- x³-12*x=0
- x to the power of 3-12*x=0
- x^3-12x=0
- x3-12x=0
- x^3-12*x=O
-
Similar expressions
- x^3+12*x=0
x^3-12*x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$x^{3} - 12 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - 12\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - 12 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = -12$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{2} = 2 \sqrt{3}$$
$$x_{3} = - 2 \sqrt{3}$$
The final answer for x^3 - 12*x = 0:
$$x_{1} = 0$$
$$x_{2} = 2 \sqrt{3}$$
$$x_{3} = - 2 \sqrt{3}$$
$$x^{3} - 12 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - 12\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - 12 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = -12$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-12) = 48
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = 2 \sqrt{3}$$
$$x_{3} = - 2 \sqrt{3}$$
The final answer for x^3 - 12*x = 0:
$$x_{1} = 0$$
$$x_{2} = 2 \sqrt{3}$$
$$x_{3} = - 2 \sqrt{3}$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -12$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -12$$
$$x_{1} x_{2} x_{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -12$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -12$$
$$x_{1} x_{2} x_{3} = 0$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
___ x2 = -2*\/ 3
$$x_{2} = - 2 \sqrt{3}$$
___ x3 = 2*\/ 3
$$x_{3} = 2 \sqrt{3}$$
x3 = 2*sqrt(3)
Sum and product of roots
[src]
sum
___ ___ - 2*\/ 3 + 2*\/ 3
$$- 2 \sqrt{3} + 2 \sqrt{3}$$
=
0
$$0$$
product
___ ___ 0*-2*\/ 3 *2*\/ 3
$$2 \sqrt{3} \cdot 0 \left(- 2 \sqrt{3}\right)$$
=
0
$$0$$
0
The graph