Mister Exam
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How to use it?
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Identical expressions
- x^ three + nine *x= zero
- x cubed plus 9 multiply by x equally 0
- x to the power of three plus nine multiply by x equally zero
- x3+9*x=0
- x³+9*x=0
- x to the power of 3+9*x=0
- x^3+9x=0
- x3+9x=0
- x^3+9*x=O
-
Similar expressions
- x^3-9*x=0
x^3+9*x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$x^{3} + 9 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} + 9\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} + 9 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = 9$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = 3 i$$
$$x_{3} = - 3 i$$
The final answer for x^3 + 9*x = 0:
$$x_{1} = 0$$
$$x_{2} = 3 i$$
$$x_{3} = - 3 i$$
$$x^{3} + 9 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} + 9\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} + 9 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = 9$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (9) = -36
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
$$x_{2} = 3 i$$
$$x_{3} = - 3 i$$
The final answer for x^3 + 9*x = 0:
$$x_{1} = 0$$
$$x_{2} = 3 i$$
$$x_{3} = - 3 i$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 9$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 9$$
$$x_{1} x_{2} x_{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 9$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 9$$
$$x_{1} x_{2} x_{3} = 0$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
x2 = -3*I
$$x_{2} = - 3 i$$
x3 = 3*I
$$x_{3} = 3 i$$
x3 = 3*i
Sum and product of roots
[src]
sum
-3*I + 3*I
$$- 3 i + 3 i$$
=
0
$$0$$
product
0*-3*I*3*I
$$3 i 0 \left(- 3 i\right)$$
=
0
$$0$$
0
The graph