Mister Exam
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How to use it?
- Equation:
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Equation -x^6-x^3+2=0
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Identical expressions
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Similar expressions
- -x^6-x^3-2=0
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- x^6-x^3+2=0
-x^6-x^3+2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$- x^{6} - x^{3} + 2 = 0$$
Do replacement
$$v = x^{3}$$
then the equation will be the:
$$- v^{2} - v + 2 = 0$$
This equation is of the form
$$a*v^2 + b*v + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = -1$$
$$b = -1$$
$$c = 2$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right)^{2} - \left(-1\right) 4 \cdot 2 = 9$$
Because D > 0, then the equation has two roots.
$$v_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$v_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$v_{1} = -2$$
Simplify
$$v_{2} = 1$$
Simplify
The final answer:
Because
$$v = x^{3}$$
then
$$x_{1} = \sqrt[3]{v_{1}}$$
$$x_{3} = \sqrt[3]{v_{2}}$$
then:
$$x_{1} = \frac{0}{1} + \frac{1 \left(-2\right)^{\frac{1}{3}}}{1} = \sqrt[3]{-2}$$
$$x_{3} = \frac{0}{1} + \frac{1 \cdot 1^{\frac{1}{3}}}{1} = 1$$
$$- x^{6} - x^{3} + 2 = 0$$
Do replacement
$$v = x^{3}$$
then the equation will be the:
$$- v^{2} - v + 2 = 0$$
This equation is of the form
$$a*v^2 + b*v + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = -1$$
$$b = -1$$
$$c = 2$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right)^{2} - \left(-1\right) 4 \cdot 2 = 9$$
Because D > 0, then the equation has two roots.
$$v_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$v_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$v_{1} = -2$$
Simplify
$$v_{2} = 1$$
Simplify
The final answer:
Because
$$v = x^{3}$$
then
$$x_{1} = \sqrt[3]{v_{1}}$$
$$x_{3} = \sqrt[3]{v_{2}}$$
then:
$$x_{1} = \frac{0}{1} + \frac{1 \left(-2\right)^{\frac{1}{3}}}{1} = \sqrt[3]{-2}$$
$$x_{3} = \frac{0}{1} + \frac{1 \cdot 1^{\frac{1}{3}}}{1} = 1$$
Rapid solution
[src]
x_1 = 1
$$x_{1} = 1$$
3 ___ x_2 = -\/ 2
$$x_{2} = - \sqrt[3]{2}$$
___
1 I*\/ 3
x_3 = - - - -------
2 2
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
___
1 I*\/ 3
x_4 = - - + -------
2 2
$$x_{4} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
3 ___ 3 ___ ___
\/ 2 I*\/ 2 *\/ 3
x_5 = ----- - -------------
2 2
$$x_{5} = \frac{\sqrt[3]{2}}{2} - \frac{\sqrt[3]{2} \sqrt{3} i}{2}$$
3 ___ 3 ___ ___
\/ 2 I*\/ 2 *\/ 3
x_6 = ----- + -------------
2 2
$$x_{6} = \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{2} \sqrt{3} i}{2}$$
Sum and product of roots
[src]
sum
___ ___ 3 ___ 3 ___ ___ 3 ___ 3 ___ ___
3 ___ 1 I*\/ 3 1 I*\/ 3 \/ 2 I*\/ 2 *\/ 3 \/ 2 I*\/ 2 *\/ 3
1 + -\/ 2 + - - - ------- + - - + ------- + ----- - ------------- + ----- + -------------
2 2 2 2 2 2 2 2
$$\left(1\right) + \left(- \sqrt[3]{2}\right) + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) + \left(\frac{\sqrt[3]{2}}{2} - \frac{\sqrt[3]{2} \sqrt{3} i}{2}\right) + \left(\frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{2} \sqrt{3} i}{2}\right)$$
=
0
$$0$$
product
___ ___ 3 ___ 3 ___ ___ 3 ___ 3 ___ ___
3 ___ 1 I*\/ 3 1 I*\/ 3 \/ 2 I*\/ 2 *\/ 3 \/ 2 I*\/ 2 *\/ 3
1 * -\/ 2 * - - - ------- * - - + ------- * ----- - ------------- * ----- + -------------
2 2 2 2 2 2 2 2
$$\left(1\right) * \left(- \sqrt[3]{2}\right) * \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) * \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) * \left(\frac{\sqrt[3]{2}}{2} - \frac{\sqrt[3]{2} \sqrt{3} i}{2}\right) * \left(\frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{2} \sqrt{3} i}{2}\right)$$
=
-2
$$-2$$
Numerical answer
[src]
x1 = 0.629960524947437 - 1.09112363597172*i
x2 = 1.0
x3 = -0.5 - 0.866025403784439*i
x4 = -0.5 + 0.866025403784439*i
x5 = 0.629960524947437 + 1.09112363597172*i
x6 = -1.25992104989487
x6 = -1.25992104989487
The graph