(6)/(x-3)=(8x^(2))/(x^(2)-9)-(4x)/(x+3)

4 min read Jun 16, 2024

Solving the Equation: (6)/(x-3)=(8x^(2))/(x^(2)-9)-(4x)/(x+3)

This article will guide you through the process of solving the given equation:

(6)/(x-3)=(8x^(2))/(x^(2)-9)-(4x)/(x+3)

1. Factor the denominator:

The first step is to factor the denominators to identify any common factors and simplify the equation. Notice that (x^2 - 9) is a difference of squares and can be factored as (x+3)(x-3).

Our equation now becomes:

(6)/(x-3)=(8x^(2))/((x+3)(x-3))-(4x)/(x+3)

2. Find a common denominator:

The least common denominator (LCD) for all fractions is (x+3)(x-3). We need to multiply each fraction by the appropriate factor to achieve this common denominator.

For the first fraction (6)/(x-3): We need to multiply both numerator and denominator by (x+3).
For the second fraction (8x^(2))/((x+3)(x-3)): It already has the common denominator.
For the third fraction (4x)/(x+3): We need to multiply both numerator and denominator by (x-3).

This gives us:

(6(x+3))/((x+3)(x-3))=(8x^(2))/((x+3)(x-3))-(4x(x-3))/((x+3)(x-3))

3. Combine the fractions:

Now that all fractions have the same denominator, we can combine them:

(6(x+3)-8x^(2)+4x(x-3))/((x+3)(x-3)) = 0

4. Simplify the equation:

Expand and simplify the numerator:

(6x + 18 - 8x^2 + 4x^2 - 12x)/((x+3)(x-3)) = 0

(-4x^2 - 6x + 18)/((x+3)(x-3)) = 0

5. Solve for x:

Since the fraction equals zero, the numerator must equal zero:

-4x^2 - 6x + 18 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Where a = -4, b = -6, and c = 18.

Substituting these values into the quadratic formula and simplifying, we get:

x = (6 ± √(36 + 288)) / (-8)

x = (6 ± √324) / (-8)

x = (6 ± 18) / (-8)

This gives us two solutions:

x = -3 and x = 3/2

6. Check for extraneous solutions:

Remember that our original equation had denominators that could equal zero for certain values of x. We need to check if our solutions make any of the denominators zero.

x = -3 would make the denominator (x+3) equal zero. Therefore, x = -3 is an extraneous solution and must be discarded.
x = 3/2 does not make any of the denominators zero. Therefore, it is a valid solution.

Conclusion:

The only valid solution to the equation (6)/(x-3)=(8x^(2))/(x^(2)-9)-(4x)/(x+3) is x = 3/2.