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

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

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

Combine like terms: 29x²  30x = 2

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

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.