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

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

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

This article will guide you through the steps of solving the given equation: (5x - 3)² + (2x - 3)(2x + 3) = 2.

Understanding the Equation

The equation involves:

  • Squaring a binomial: (5x - 3)²
  • Multiplying conjugates: (2x - 3)(2x + 3)

We'll use the following algebraic identities to simplify:

  • (a - b)² = a² - 2ab + b²
  • (a - b)(a + b) = a² - b²

Steps to Solve

  1. Expand the squares and the product of conjugates: (5x - 3)² = (5x)² - 2(5x)(3) + 3² = 25x² - 30x + 9 (2x - 3)(2x + 3) = (2x)² - 3² = 4x² - 9

  2. Substitute the expanded terms back into the equation: 25x² - 30x + 9 + 4x² - 9 = 2

  3. Combine like terms: 29x² - 30x = 2

  4. Move all terms to one side to form a quadratic equation: 29x² - 30x - 2 = 0

  5. Solve the quadratic equation: This equation can be solved using the quadratic formula:

    x = (-b ± √(b² - 4ac)) / 2a

    Where:

    • a = 29
    • b = -30
    • c = -2

    Substitute the values and solve for x.

Conclusion

By following these steps, you can solve the equation (5x - 3)² + (2x - 3)(2x + 3) = 2. Remember to carefully apply the algebraic identities and simplify the equation to arrive at the solution(s) for x.

Featured Posts