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(4x-5)^2-(2x+3)^2=0

2 min read Jun 16, 2024
(4x-5)^2-(2x+3)^2=0

Solving the Equation (4x-5)^2 - (2x+3)^2 = 0

This equation involves the difference of squares, which can be factored to simplify the process. Here's how to solve it:

Factoring the Equation

  1. Recognize the Difference of Squares: The equation is in the form of a² - b² = 0, where a = (4x - 5) and b = (2x + 3).

  2. Apply the Difference of Squares Formula: The difference of squares formula states that a² - b² = (a + b)(a - b). Applying this to our equation:

    [(4x - 5) + (2x + 3)][(4x - 5) - (2x + 3)] = 0

  3. Simplify the Expression:

    (6x - 2)(2x - 8) = 0

Solving for x

Now we have a product of two factors that equals zero. For this to be true, at least one of the factors must be equal to zero.

  1. Set each factor to zero:

    • 6x - 2 = 0
    • 2x - 8 = 0

  2. Solve for x in each equation:

    • 6x = 2 => x = 1/3
    • 2x = 8 => x = 4

Solution

Therefore, the solutions to the equation (4x - 5)² - (2x + 3)² = 0 are x = 1/3 and x = 4.

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