(4x-3)(2x-1)=(x-3)(4x-3)

3 min read Jun 16, 2024

Solving the Equation: (4x-3)(2x-1) = (x-3)(4x-3)

This equation presents a classic example of solving a quadratic equation through simplification and factorization. Let's break down the steps:

1. Expanding the Products

First, we need to expand the products on both sides of the equation using the distributive property (also known as FOIL):

Left side: (4x - 3)(2x - 1) = 8x² - 4x - 6x + 3 = 8x² - 10x + 3
Right side: (x - 3)(4x - 3) = 4x² - 3x - 12x + 9 = 4x² - 15x + 9

Now, the equation becomes: 8x² - 10x + 3 = 4x² - 15x + 9

2. Combining Like Terms

To simplify the equation further, let's move all terms to one side:

Subtract 4x² from both sides: 4x² - 10x + 3 = -15x + 9
Add 15x to both sides: 4x² + 5x + 3 = 9
Subtract 9 from both sides: 4x² + 5x - 6 = 0

Now, we have a standard quadratic equation: 4x² + 5x - 6 = 0

3. Factoring the Quadratic Equation

The next step is to factor the quadratic equation. We need to find two numbers that add up to 5 (the coefficient of the x term) and multiply to -24 (the product of the coefficient of the x² term and the constant term).

The numbers 8 and -3 satisfy these conditions: 8 + (-3) = 5 and 8 * (-3) = -24.

Now, we can rewrite the equation as:

(4x - 3)(x + 2) = 0

4. Solving for x

For the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:

4x - 3 = 0
- Add 3 to both sides: 4x = 3
- Divide both sides by 4: x = 3/4
x + 2 = 0
- Subtract 2 from both sides: x = -2

5. Conclusion

Therefore, the solutions to the equation (4x - 3)(2x - 1) = (x - 3)(4x - 3) are x = 3/4 and x = -2.