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(3x-2)(2x-3)=(2x+5)(2x-1)

3 min read Jun 16, 2024
(3x-2)(2x-3)=(2x+5)(2x-1)

Solving the Equation: (3x-2)(2x-3) = (2x+5)(2x-1)

This article will guide you through the steps to solve the equation (3x-2)(2x-3) = (2x+5)(2x-1).

Expanding the Equation

First, we need to expand both sides of the equation using the FOIL method (First, Outer, Inner, Last):

  • Left Side:
    • (3x-2)(2x-3) = (3x * 2x) + (3x * -3) + (-2 * 2x) + (-2 * -3)
    • = 6x² - 9x - 4x + 6
    • = 6x² - 13x + 6
  • Right Side:
    • (2x+5)(2x-1) = (2x * 2x) + (2x * -1) + (5 * 2x) + (5 * -1)
    • = 4x² - 2x + 10x - 5
    • = 4x² + 8x - 5

Simplifying the Equation

Now, our equation looks like this: 6x² - 13x + 6 = 4x² + 8x - 5

To simplify, we need to move all terms to one side: 6x² - 13x + 6 - 4x² - 8x + 5 = 0

Combining like terms: 2x² - 21x + 11 = 0

Solving the Quadratic Equation

We now have a quadratic equation. There are several ways to solve it:

  • Factoring: Attempt to factor the quadratic expression. If it can be factored, it will lead to two linear equations.
  • Quadratic Formula: The most general approach, which always works:
    • x = [-b ± √(b² - 4ac)] / 2a
    • In our equation, a = 2, b = -21, and c = 11.

Let's use the quadratic formula:

  1. Substitute the values: x = [21 ± √((-21)² - 4 * 2 * 11)] / (2 * 2)

  2. Simplify: x = [21 ± √(441 - 88)] / 4 x = [21 ± √353] / 4

  3. Calculate the two solutions: x1 = (21 + √353) / 4 x2 = (21 - √353) / 4

Solutions

Therefore, the solutions to the equation (3x-2)(2x-3) = (2x+5)(2x-1) are:

  • x = (21 + √353) / 4
  • x = (21 - √353) / 4

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