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Equation 7x^3-28x=0
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Identical expressions
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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
$$a\ x^2 + b\ x + c = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 7$$
$$b = 0$$
$$c = -28$$
, then
$$D = b^2 - 4\ a\ c = $$
$$0^{2} - 7 \cdot 4 \left(-28\right) = 784$$
Because D > 0, then the equation has two roots.
$$x_2 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_3 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{2} = 2$$
Simplify
$$x_{3} = -2$$
Simplify
The final answer for (7*x^3 - 28*x) + 0 = 0:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{3} = -2$$
$$7 x^{3} - 28 x = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 7$$
$$b = 0$$
$$c = -28$$
, then
$$D = b^2 - 4\ a\ c = $$
$$0^{2} - 7 \cdot 4 \left(-28\right) = 784$$
Because D > 0, then the equation has two roots.
$$x_2 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_3 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{2} = 2$$
Simplify
$$x_{3} = -2$$
Simplify
The final answer for (7*x^3 - 28*x) + 0 = 0:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{3} = -2$$
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
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
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
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -4$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -4$$
$$x_{1} x_{2} x_{3} = 0$$
Sum and product of roots
[src]
sum
-2 + 0 + 2
$$\left(-2\right) + \left(0\right) + \left(2\right)$$
=
0
$$0$$
product
-2 * 0 * 2
$$\left(-2\right) * \left(0\right) * \left(2\right)$$
=
0
$$0$$
The graph