0 = 2x^3 + x - 3
solve for x
x = 0
how? can you show me please?
No matter what is on the other side of the equation, it still says that it equals zero.
but if you move things around it could equal something else...
like 3 = 2x^3 + x
how do you solve from there?
To solve the equation 0 = 2x^3 + x - 3 for x, we can use the method of factoring or the method of numerical approximation. Let's start with factoring:
1. Rearrange the equation and set it equal to zero:
2x^3 + x - 3 = 0
2. Factor out common factors, if possible. In this case, there is no common factor.
3. Try to express the remaining polynomial as a product of binomials or trinomials.
Looking at the expression, we see it is a cubic polynomial with no obvious factorization. Therefore, we will use numerical approximation methods to find its approximate solutions.
One common numerical method is the Newton-Raphson method. Here's how you can use it:
1. Take the derivative of the polynomial to find its slope at any given x-value. The derivative of the given polynomial is: f'(x) = 6x^2 + 1.
2. Choose an initial guess for the root. Let's start with x = 1.
3. Use the formula: x1 = x0 - (f(x0) / f'(x0)) to find a better approximation. Substituting the values, we have:
x1 = 1 - (2(1)^3 + 1 - 3) / (6(1)^2 + 1)
= 1 - (2 + 1 - 3) / (6 + 1)
= 1 - 0 / 7
= 1
4. Repeat the previous step with the new approximation until you get a desired level of accuracy. In this case, if we substitute x1 = 1 back into the formula, we would get x2 = 1 as well.
5. Keep iterating this process until the difference between two consecutive approximations is small enough (e.g., less than 0.0001). In this case, the equation has only one real root near x = 1.
Therefore, the solution to the equation 0 = 2x^3 + x - 3 is approximately x = 1.