(x+3/x-2)^2+6(x-3/x+2)^2-7(x^2-9)/x^2-4=0

4 min read Jun 16, 2024

Solving the Equation: (x+3/x-2)^2+6(x-3/x+2)^2-7(x^2-9)/x^2-4=0

This equation presents a complex challenge involving rational expressions and quadratic terms. To solve it, we will employ a combination of algebraic manipulation, simplification, and factoring.

1. Simplifying the Equation:

First, we can simplify the equation by recognizing that the denominators of some of the terms are similar:

x² - 4 can be factored as (x + 2)(x - 2)

Now, let's rewrite the equation with these factored terms:

(x+3/x-2)² + 6(x-3/x+2)² - 7(x²-9)/(x+2)(x-2) = 0

2. Finding a Common Denominator:

To combine the terms effectively, we need to find a common denominator for all fractions. The least common multiple of the denominators is (x+2)²(x-2)².

Let's rewrite each term with this common denominator:

(x+3/x-2)² = (x+3)²(x+2)² / (x-2)²(x+2)²
6(x-3/x+2)² = 6(x-3)²(x-2)² / (x+2)²(x-2)²
7(x²-9)/(x+2)(x-2) = 7(x²-9)(x+2)(x-2) / (x+2)²(x-2)²

Now we have:

(x+3)²(x+2)² / (x-2)²(x+2)² + 6(x-3)²(x-2)² / (x+2)²(x-2)² - 7(x²-9)(x+2)(x-2) / (x+2)²(x-2)² = 0

3. Combining Terms:

Since all terms now share the same denominator, we can combine their numerators:

[(x+3)²(x+2)² + 6(x-3)²(x-2)² - 7(x²-9)(x+2)(x-2)] / (x+2)²(x-2)² = 0

4. Expanding and Simplifying:

Let's expand the squares and multiply the terms in the numerator:

[x⁴ + 8x³ + 13x² + 12x + 9 + 6x⁴ - 72x³ + 174x² - 216x + 324 - 7x⁴ + 63x² - 126] / (x+2)²(x-2)² = 0

Now, combine like terms:

[0x⁴ - 64x³ + 250x² - 204x + 213] / (x+2)²(x-2)² = 0

5. Solving for x:

For the entire expression to equal zero, the numerator must be equal to zero:

-64x³ + 250x² - 204x + 213 = 0

Unfortunately, this cubic equation doesn't have an easily factored solution. You can use numerical methods like the Newton-Raphson method or utilize a graphing calculator to find approximate solutions for x.

Important Note: While we have simplified the equation and reduced it to a cubic equation, solving for x directly is still a complex task. You would need to employ advanced numerical techniques to obtain solutions.