Mister Exam
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- Equation:
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Identical expressions
- x^ three + two *x- three = zero
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Similar expressions
- x^3-2*x-3=0
- x^3+2*x+3=0
x^3+2*x-3=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(x^{3} + 2 x\right) - 3 = 0$$
transform
$$\left(2 x + \left(x^{3} - 1\right)\right) - 2 = 0$$
or
$$\left(2 x + \left(x^{3} - 1^{3}\right)\right) - 2 = 0$$
$$2 \left(x - 1\right) + \left(x^{3} - 1^{3}\right) = 0$$
$$\left(x - 1\right) \left(\left(x^{2} + x\right) + 1^{2}\right) + 2 \left(x - 1\right) = 0$$
Take common factor -1 + x from the equation
we get:
$$\left(x - 1\right) \left(\left(\left(x^{2} + x\right) + 1^{2}\right) + 2\right) = 0$$
or
$$\left(x - 1\right) \left(x^{2} + x + 3\right) = 0$$
then:
$$x_{1} = 1$$
and also
we get the equation
$$x^{2} + x + 3 = 0$$
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
$$a = 1$$
$$b = 1$$
$$c = 3$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
The final answer for x^3 + 2*x - 3 = 0:
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
$$\left(x^{3} + 2 x\right) - 3 = 0$$
transform
$$\left(2 x + \left(x^{3} - 1\right)\right) - 2 = 0$$
or
$$\left(2 x + \left(x^{3} - 1^{3}\right)\right) - 2 = 0$$
$$2 \left(x - 1\right) + \left(x^{3} - 1^{3}\right) = 0$$
$$\left(x - 1\right) \left(\left(x^{2} + x\right) + 1^{2}\right) + 2 \left(x - 1\right) = 0$$
Take common factor -1 + x from the equation
we get:
$$\left(x - 1\right) \left(\left(\left(x^{2} + x\right) + 1^{2}\right) + 2\right) = 0$$
or
$$\left(x - 1\right) \left(x^{2} + x + 3\right) = 0$$
then:
$$x_{1} = 1$$
and also
we get the equation
$$x^{2} + x + 3 = 0$$
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
$$a = 1$$
$$b = 1$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (3) = -11
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
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
The final answer for x^3 + 2*x - 3 = 0:
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -3$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -3$$
Sum and product of roots
[src]
sum
____ ____
1 I*\/ 11 1 I*\/ 11
1 + - - - -------- + - - + --------
2 2 2 2
$$\left(1 + \left(- \frac{1}{2} - \frac{\sqrt{11} i}{2}\right)\right) + \left(- \frac{1}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
0
$$0$$
product
/ ____\ / ____\ | 1 I*\/ 11 | | 1 I*\/ 11 | |- - - --------|*|- - + --------| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} - \frac{\sqrt{11} i}{2}\right) \left(- \frac{1}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
3
$$3$$
3
Rapid solution
[src]
x1 = 1
$$x_{1} = 1$$
____
1 I*\/ 11
x2 = - - - --------
2 2
$$x_{2} = - \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
____
1 I*\/ 11
x3 = - - + --------
2 2
$$x_{3} = - \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
x3 = -1/2 + sqrt(11)*i/2
Numerical answer
[src]
x1 = 1.0
x2 = -0.5 + 1.6583123951777*i
x3 = -0.5 - 1.6583123951777*i
x3 = -0.5 - 1.6583123951777*i
The graph