What does 3x^3-3x+2=0 come out to? How would I solve this?
Is the x cubed supposed to be x squared?
You can compare the equation with the identity:
(a+b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3
which you can rewrite as:
(a + b)^3 = 3 ab (a + b) + a^3 + b^3 (1)
The equation you want to solve is:
x^3 = x - 2 (2)
Equation (1) is an identity that is always true. I.e., whatever you substitute for a and b, the equation is always satisfied. Equation (2) has, of course, at most three solutions. Suppose you choose a and b such that:
3 ab = 1 (3)
and
a^3 + b^3 = -2 (4)
then equation (1) says that:
(a + b)^3 = (a + b) - 2
and you see that a + b would then satisfy equation (2). So, how do we solve equations (3) and (4) for a and b? You take the cube of equation (3):
3 ab = 1 --->
a^3 b^3 = 1/27
If you put A = a^3 and B = b^3, then:
A B = 1/27 (5)
equation (4) can be written as:
A + B = -2 (6)
If you use (6) to express B in terms of A and substitute that in (5) yo obtain a quadratic equation for A. Solve that equation. You obtain two solutions. Because of the symmetry between A and B (i.e. if you interchange A and B in (5) and (6) the two equatons remain the same), you can take A to be one of the solutions and B to be the other solution.
Extract the cube roots to obtain a and b, add them up to obtain the solution. To obtain the two other solutions, you must use that for given A there are three solutions for a, such that
a^3 = A:
a = A^(1/3) Exp[2 n pi i/3]
where n is 0, 1 or 2. The same is true for b, but we need to satisfy
equation (3). This means that if we choose a to be:
a = A^(1/3) Exp[2 n pi i/3]
then b is given by:
b = B^(1/3) Exp[-2 n pi i/3]
To solve the equation 3x^3 - 3x + 2 = 0, you can use a numerical method like Newton's method or the bisection method. These methods involve iteratively refining an initial guess until you find a solution that satisfies the equation.
Alternatively, you can use a symbolic solver or graphing calculator to find the exact solutions. You can rewrite the equation as:
x^3 - x + (2/3) = 0
Unfortunately, there is no simple algebraic method to solve this equation for x. The solutions involve complex numbers and are not expressible in terms of elementary functions. Therefore, using a numerical method or a calculator is the most practical approach to find the solutions.
If the equation you meant is 3x^2 - 3x + 2 = 0, then you can use the quadratic formula to find the solutions. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 3x^2 - 3x + 2 = 0, a = 3, b = -3, and c = 2. Plugging in these values into the quadratic formula, you can find the solutions for x.