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

4 min read Jun 16, 2024

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

This equation involves a fascinating interplay of algebraic manipulation and the concept of zero product property. Let's break down the steps to solve for the value(s) of 'x'.

1. Simplify and Rearrange

Firstly, we need to simplify the equation by expanding the squares and rearranging terms.

Expand: Use the FOIL method (First, Outer, Inner, Last) to expand the squared terms.
- (2x-3)^2 = (2x-3)(2x-3) = 4x^2 - 12x + 9
- (4x+5)^2 = (4x+5)(4x+5) = 16x^2 + 40x + 25
Substitute: Substitute the expanded forms back into the original equation.
- (4x^2 - 12x + 9)(4x+5) = (2x-3)(16x^2 + 40x + 25)
Distribute: Distribute the terms on both sides of the equation.
- 16x^3 + 20x^2 - 48x^2 - 60x + 36x + 45 = 32x^3 + 80x^2 + 50x - 48x^2 - 120x - 75
Combine like terms: Combine the terms on each side.
- 16x^3 - 28x^2 - 24x + 45 = 32x^3 + 32x^2 - 70x - 75
Rearrange: Move all terms to one side to set the equation equal to zero.
- 0 = 16x^3 + 32x^2 - 28x^2 + 32x - 24x + 75 - 45
- 0 = 16x^3 + 4x^2 + 8x + 30

2. Factor the Equation

The next step is to factor the equation to find the possible solutions. This can be tricky, but we can use a few techniques:

Factoring by grouping: While not always effective, you can try to group terms and see if you can extract common factors.
Rational Root Theorem: This theorem helps identify potential rational roots of a polynomial. It states that any rational root of the polynomial must be a factor of the constant term (30) divided by a factor of the leading coefficient (16). This gives you a list of potential rational roots to test using synthetic division or by substitution.

3. Applying the Zero Product Property

Once you have factored the equation, the Zero Product Property comes into play. This property states that if the product of two or more factors is zero, at least one of the factors must be zero.

Example: If you factor the equation into (x-a)(x-b)(x-c) = 0, then either (x-a) = 0, or (x-b) = 0, or (x-c) = 0.

4. Solve for 'x'

By applying the Zero Product Property and solving each resulting equation, you can find the values of 'x' that satisfy the original equation.

Important Note: Factoring this equation might be challenging due to its degree. It's possible that the equation has no real solutions, or that it requires more advanced factoring techniques. You may need to use numerical methods or graphing tools to approximate solutions.