Mister Exam
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Identical expressions
- 7x^ three +28x= zero
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Similar expressions
- 7x^3-28x=0
7x^3+28x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$7 x^{3} + 28 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(7 x^{2} + 28\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$7 x^{2} + 28 = 0$$
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
$$a = 7$$
$$b = 0$$
$$c = 28$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = 2 i$$
$$x_{3} = - 2 i$$
The final answer for 7*x^3 + 28*x = 0:
$$x_{1} = 0$$
$$x_{2} = 2 i$$
$$x_{3} = - 2 i$$
$$7 x^{3} + 28 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(7 x^{2} + 28\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$7 x^{2} + 28 = 0$$
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
$$a = 7$$
$$b = 0$$
$$c = 28$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (7) * (28) = -784
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
$$x_{2} = 2 i$$
$$x_{3} = - 2 i$$
The final answer for 7*x^3 + 28*x = 0:
$$x_{1} = 0$$
$$x_{2} = 2 i$$
$$x_{3} = - 2 i$$
Vieta's Theorem
rewrite the equation
$$7 x^{3} + 28 x = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 4 x = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 4$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 4$$
$$x_{1} x_{2} x_{3} = 0$$
$$7 x^{3} + 28 x = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 4 x = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 4$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 4$$
$$x_{1} x_{2} x_{3} = 0$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
x2 = -2*I
$$x_{2} = - 2 i$$
x3 = 2*I
$$x_{3} = 2 i$$
x3 = 2*i
Sum and product of roots
[src]
sum
-2*I + 2*I
$$- 2 i + 2 i$$
=
0
$$0$$
product
0*-2*I*2*I
$$2 i 0 \left(- 2 i\right)$$
=
0
$$0$$
0