Mister Exam
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How to use it?
- Equation:
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Equation x³-2x²+2x-4=0
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Equation (x+5)(2x-4)+24=0
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Equation 3y^2
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Identical expressions
- x³-2x²+2x- four = zero
- x³ minus 2x² plus 2x minus 4 equally 0
- x³ minus 2x² plus 2x minus four equally zero
- x³-2x²+2x-4=O
-
Similar expressions
- x³+2x²+2x-4=0
- x³-2x²-2x-4=0
- x³-2x²+2x+4=0
x³-2x²+2x-4=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$x^{3} - 2 x^{2} + 2 x - 4 = 0$$
transform
$$x^{3} - 2 x^{2} + 2 x - 4 = 0$$
or
$$x^{3} + 2 x - 12 = 0$$
$$x^{3} - 2 x^{2} + 2 x - 4 = 0$$
$$\left(- 2 x + 4\right) \left(x + 2\right) + \left(x - 2\right) \left(x^{2} + 2 x + 4\right) + 2 x - 4 = 0$$
Take common factor $x - 2$ from the equation
we get:
$$\left(x - 2\right) \left(x^{2} + 2\right) = 0$$
or
$$\left(x - 2\right) \left(x^{2} + 2\right) = 0$$
then:
$$x_{1} = 2$$
and also
we get the equation
$$x^{2} + 2 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = 2$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 2 + 0^{2} = -8$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$x_2 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_3 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{2} = \sqrt{2} i$$
Simplify
$$x_{3} = - \sqrt{2} i$$
Simplify
The final answer for (x^3 - 2*x^2 + 2*x - 1*4) + 0 = 0:
$$x_{1} = 2$$
$$x_{2} = \sqrt{2} i$$
$$x_{3} = - \sqrt{2} i$$
$$x^{3} - 2 x^{2} + 2 x - 4 = 0$$
transform
$$x^{3} - 2 x^{2} + 2 x - 4 = 0$$
or
$$x^{3} + 2 x - 12 = 0$$
$$x^{3} - 2 x^{2} + 2 x - 4 = 0$$
$$\left(- 2 x + 4\right) \left(x + 2\right) + \left(x - 2\right) \left(x^{2} + 2 x + 4\right) + 2 x - 4 = 0$$
Take common factor $x - 2$ from the equation
we get:
$$\left(x - 2\right) \left(x^{2} + 2\right) = 0$$
or
$$\left(x - 2\right) \left(x^{2} + 2\right) = 0$$
then:
$$x_{1} = 2$$
and also
we get the equation
$$x^{2} + 2 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = 2$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 2 + 0^{2} = -8$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$x_2 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_3 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{2} = \sqrt{2} i$$
Simplify
$$x_{3} = - \sqrt{2} i$$
Simplify
The final answer for (x^3 - 2*x^2 + 2*x - 1*4) + 0 = 0:
$$x_{1} = 2$$
$$x_{2} = \sqrt{2} i$$
$$x_{3} = - \sqrt{2} i$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -4$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -4$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -4$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -4$$
Sum and product of roots
[src]
sum
___ ___ 2 + -I*\/ 2 + I*\/ 2
$$\left(2\right) + \left(- \sqrt{2} i\right) + \left(\sqrt{2} i\right)$$
=
2
$$2$$
product
___ ___ 2 * -I*\/ 2 * I*\/ 2
$$\left(2\right) * \left(- \sqrt{2} i\right) * \left(\sqrt{2} i\right)$$
=
4
$$4$$
Rapid solution
[src]
x_1 = 2
$$x_{1} = 2$$
___ x_2 = -I*\/ 2
$$x_{2} = - \sqrt{2} i$$
___ x_3 = I*\/ 2
$$x_{3} = \sqrt{2} i$$
Numerical answer
[src]
x1 = -1.4142135623731*i
x2 = 2.0
x3 = 1.4142135623731*i
x3 = 1.4142135623731*i
The graph