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

Jasonbradley

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Jun 16, 2024

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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.