Mister Exam

Other calculators


x^3-3*x^2+2=0

x^3-3*x^2+2=0 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src]
 3      2        
x  - 3*x  + 2 = 0
$$\left(x^{3} - 3 x^{2}\right) + 2 = 0$$
Detail solution
Given the equation:
$$\left(x^{3} - 3 x^{2}\right) + 2 = 0$$
transform
$$\left(- 3 x^{2} + \left(x^{3} - 1\right)\right) + 3 = 0$$
or
$$\left(- 3 x^{2} + \left(x^{3} - 1^{3}\right)\right) + 3 \cdot 1^{2} = 0$$
$$- 3 \left(x^{2} - 1^{2}\right) + \left(x^{3} - 1^{3}\right) = 0$$
$$- 3 \left(x - 1\right) \left(x + 1\right) + \left(x - 1\right) \left(\left(x^{2} + x\right) + 1^{2}\right) = 0$$
Take common factor -1 + x from the equation
we get:
$$\left(x - 1\right) \left(- 3 \left(x + 1\right) + \left(\left(x^{2} + x\right) + 1^{2}\right)\right) = 0$$
or
$$\left(x - 1\right) \left(x^{2} - 2 x - 2\right) = 0$$
then:
$$x_{1} = 1$$
and also
we get the equation
$$x^{2} - 2 x - 2 = 0$$
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 - it is the discriminant.
Because
$$a = 1$$
$$b = -2$$
$$c = -2$$
, then
D = b^2 - 4 * a * c = 

(-2)^2 - 4 * (1) * (-2) = 12

Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)

x3 = (-b - sqrt(D)) / (2*a)

or
$$x_{2} = 1 + \sqrt{3}$$
$$x_{3} = 1 - \sqrt{3}$$
The final answer for x^3 - 3*x^2 + 2 = 0:
$$x_{1} = 1$$
$$x_{2} = 1 + \sqrt{3}$$
$$x_{3} = 1 - \sqrt{3}$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 2$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 2$$
The graph
Rapid solution [src]
x1 = 1
$$x_{1} = 1$$
           ___
x2 = 1 - \/ 3 
$$x_{2} = 1 - \sqrt{3}$$
           ___
x3 = 1 + \/ 3 
$$x_{3} = 1 + \sqrt{3}$$
x3 = 1 + sqrt(3)
Sum and product of roots [src]
sum
          ___         ___
1 + 1 - \/ 3  + 1 + \/ 3 
$$\left(\left(1 - \sqrt{3}\right) + 1\right) + \left(1 + \sqrt{3}\right)$$
=
3
$$3$$
product
/      ___\ /      ___\
\1 - \/ 3 /*\1 + \/ 3 /
$$\left(1 - \sqrt{3}\right) \left(1 + \sqrt{3}\right)$$
=
-2
$$-2$$
-2
Numerical answer [src]
x1 = 1.0
x2 = 2.73205080756888
x3 = -0.732050807568877
x3 = -0.732050807568877
The graph
x^3-3*x^2+2=0 equation