Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation x^4-8x^2-9=0
- Equation x^4-15x^2-16=0
-
Equation x^3-3x^2+2x=210
-
Equation 29+y*4=309
- Express {x} in terms of y where:
- 15*x-15*y=16
- 9*x+18*y=6
- (x^2+y^2-1)^3+x^2*y^3=0
- -1*x-19*y=-10
- Graphing y =:
-
x^3-3x^2+2x
- Integral of d{x}:
-
x^3-3x^2+2x
-
Identical expressions
- x^ three -3x^ two +2x= two hundred and ten
- x cubed minus 3x squared plus 2x equally 210
- x to the power of three minus 3x to the power of two plus 2x equally two hundred and ten
- x3-3x2+2x=210
- x³-3x²+2x=210
- x to the power of 3-3x to the power of 2+2x=210
-
Similar expressions
- x^3-3x^2-2x=210
- 2^10*x2-8*x-23+2^5*x2-4*x-12-3=0
- x^3+3x^2+2x=210
- -0,1055=(2*8,314*298/96485)*arsinh(-12*10^(-2)/(2*(2*8,314*298*x*8,85*10^(-12))^(0,5)*(50)^(0,5)))
x^3-3x^2+2x=210 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$x^{3} - 3 x^{2} + 2 x = 210$$
transform
$$x^{3} - 3 x^{2} + 2 x - 210 = 0$$
or
$$x^{3} + 2 x - 357 = 0$$
$$x^{3} - 3 x^{2} + 2 x - 210 = 0$$
$$\left(- 3 x + 21\right) \left(x + 7\right) + \left(x - 7\right) \left(x^{2} + 7 x + 49\right) + 2 x - 14 = 0$$
Take common factor $x - 7$ from the equation
we get:
$$\left(x - 7\right) \left(x^{2} + 4 x + 30\right) = 0$$
or
$$\left(x - 7\right) \left(x^{2} + 4 x + 30\right) = 0$$
then:
$$x_{1} = 7$$
and also
we get the equation
$$x^{2} + 4 x + 30 = 0$$
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
$$a = 1$$
$$b = 4$$
$$c = 30$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 30 + 4^{2} = -104$$
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
$$x_{2} = -2 + \sqrt{26} i$$
Simplify
$$x_{3} = -2 - \sqrt{26} i$$
Simplify
The final answer for (x^3 - 3*x^2 + 2*x) - 210 = 0:
$$x_{1} = 7$$
$$x_{2} = -2 + \sqrt{26} i$$
$$x_{3} = -2 - \sqrt{26} i$$
$$x^{3} - 3 x^{2} + 2 x = 210$$
transform
$$x^{3} - 3 x^{2} + 2 x - 210 = 0$$
or
$$x^{3} + 2 x - 357 = 0$$
$$x^{3} - 3 x^{2} + 2 x - 210 = 0$$
$$\left(- 3 x + 21\right) \left(x + 7\right) + \left(x - 7\right) \left(x^{2} + 7 x + 49\right) + 2 x - 14 = 0$$
Take common factor $x - 7$ from the equation
we get:
$$\left(x - 7\right) \left(x^{2} + 4 x + 30\right) = 0$$
or
$$\left(x - 7\right) \left(x^{2} + 4 x + 30\right) = 0$$
then:
$$x_{1} = 7$$
and also
we get the equation
$$x^{2} + 4 x + 30 = 0$$
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
$$a = 1$$
$$b = 4$$
$$c = 30$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 30 + 4^{2} = -104$$
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
$$x_{2} = -2 + \sqrt{26} i$$
Simplify
$$x_{3} = -2 - \sqrt{26} i$$
Simplify
The final answer for (x^3 - 3*x^2 + 2*x) - 210 = 0:
$$x_{1} = 7$$
$$x_{2} = -2 + \sqrt{26} i$$
$$x_{3} = -2 - \sqrt{26} i$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -210$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -210$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -210$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -210$$
Sum and product of roots
[src]
sum
____ ____ 7 + -2 - I*\/ 26 + -2 + I*\/ 26
$$\left(7\right) + \left(-2 - \sqrt{26} i\right) + \left(-2 + \sqrt{26} i\right)$$
=
3
$$3$$
product
____ ____ 7 * -2 - I*\/ 26 * -2 + I*\/ 26
$$\left(7\right) * \left(-2 - \sqrt{26} i\right) * \left(-2 + \sqrt{26} i\right)$$
=
210
$$210$$
Rapid solution
[src]
x_1 = 7
$$x_{1} = 7$$
____ x_2 = -2 - I*\/ 26
$$x_{2} = -2 - \sqrt{26} i$$
____ x_3 = -2 + I*\/ 26
$$x_{3} = -2 + \sqrt{26} i$$
Numerical answer
[src]
x1 = 7.0
x2 = -2.0 - 5.09901951359278*i
x3 = -2.0 + 5.09901951359278*i
x3 = -2.0 + 5.09901951359278*i
The graph