(3x-4)^2/x-3 =16-24x+9x^2/15-8x+x^2

3 min read Jun 16, 2024

Solving the Equation: (3x-4)^2/x-3 = 16-24x+9x^2/15-8x+x^2

This equation involves rational expressions and requires careful manipulation to solve. Here's a step-by-step breakdown:

1. Simplify both sides of the equation

Left side: Expand the square and simplify:
- (3x-4)^2 = 9x^2 - 24x + 16
- (3x-4)^2 / (x-3) = (9x^2 - 24x + 16) / (x-3)
Right side: Factor the numerator and denominator:
- 16-24x+9x^2 = (3x-4)^2
- 15-8x+x^2 = (x-3)(x-5)
- (16-24x+9x^2) / (15-8x+x^2) = (3x-4)^2 / (x-3)(x-5)

2. Combine the terms

Now the equation becomes:

(9x^2 - 24x + 16) / (x-3) = (3x-4)^2 / (x-3)(x-5)

3. Multiply both sides by the common denominator

To get rid of the fractions, multiply both sides by (x-3)(x-5): (9x^2 - 24x + 16)(x-5) = (3x-4)^2 (x-3)

4. Expand and simplify

Expand both sides:
- 9x^3 - 69x^2 + 140x - 80 = 9x^3 - 33x^2 + 48x - 48
Combine like terms:
- -36x^2 + 92x - 32 = 0

5. Solve the quadratic equation

The simplified equation is now a quadratic equation. You can solve it using the quadratic formula:

Quadratic Formula: x = [-b ± √(b^2 - 4ac)] / 2a
Coefficients: a = -36, b = 92, c = -32
Substitute the values and solve for x.

Note: There might be multiple solutions for x. Remember to check for any extraneous solutions by substituting the obtained values back into the original equation. Extraneous solutions are values that satisfy the simplified equation but not the original equation due to division by zero.