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

3 min read Jun 17, 2024
(x-3)(x+5)=(2x+3)(x-4)

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

This article will guide you through solving the equation (x-3)(x+5) = (2x+3)(x-4). We will utilize the distributive property and algebraic manipulations to find the solution.

Expanding the Equation

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

Left Side: (x-3)(x+5) = x(x+5) - 3(x+5) = x² + 5x - 3x - 15 = x² + 2x - 15

Right Side: (2x+3)(x-4) = 2x(x-4) + 3(x-4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12

Now the equation becomes: x² + 2x - 15 = 2x² - 5x - 12

Rearranging and Simplifying

To solve for x, we need to bring all the terms to one side and simplify:

  1. Subtract x² from both sides: 2x - 15 = x² - 5x - 12

  2. Add 5x to both sides: 7x - 15 = x² - 12

  3. Add 12 to both sides: 7x - 3 = x²

  4. Subtract 7x from both sides: -3 = x² - 7x

  5. Rearrange the equation: x² - 7x + 3 = 0

Solving the Quadratic Equation

Now we have a quadratic equation in the standard form: ax² + bx + c = 0. To solve this, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, a = 1, b = -7, and c = 3. Substituting these values into the quadratic formula:

x = (7 ± √((-7)² - 4 * 1 * 3)) / (2 * 1) x = (7 ± √(49 - 12)) / 2 x = (7 ± √37) / 2

Therefore, the solutions to the equation (x-3)(x+5) = (2x+3)(x-4) are:

x = (7 + √37) / 2 and x = (7 - √37) / 2

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