((x+2)(x^(2)-2x+1))/(4+3x-x^(2)) =0.3

2 min read Jun 16, 2024

Solving the Equation: ((x+2)(x^(2)-2x+1))/(4+3x-x^(2)) = 0.3

This equation involves rational expressions and presents a good opportunity to practice algebraic manipulation. Let's break down the steps to solve it:

1. Simplify the equation

Factor the expressions:
- The numerator can be factored as (x+2)(x-1)^2.
- The denominator can be factored as (4-x)(1+x).
Rewrite the equation: The equation now becomes: ((x+2)(x-1)^2)/((4-x)(1+x)) = 0.3

2. Eliminate the denominator

Multiply both sides by the denominator:
- (x+2)(x-1)^2 = 0.3 (4-x)(1+x)

3. Expand and rearrange

Expand both sides:
- x^3 - x^2 - 3x + 2 = 1.2 - 0.3x^2 - 0.9x + 0.3x
Combine like terms:
- x^3 - 0.7x^2 - 2.1x + 1.8 = 0

4. Solve the cubic equation

Numerical methods:
- Due to the complexity of the equation, it is likely to be difficult to find an exact solution by factoring.
- You can use numerical methods like the Newton-Raphson method or graphing calculators to approximate the solution(s).

5. Check for extraneous solutions

Consider the original equation:
- Remember that the original equation involved fractions. Therefore, we must check if any solutions we find make the denominator zero.
- If any solution makes the denominator zero, it is an extraneous solution and must be discarded.

Note: There is no easy way to find the exact solutions to this cubic equation. Using numerical methods is the most practical approach.