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(2x-3)(2x+3)=(x+5)^2-9

2 min read Jun 16, 2024
(2x-3)(2x+3)=(x+5)^2-9

Solving the Equation: (2x-3)(2x+3) = (x+5)^2 - 9

This equation involves expanding products and simplifying expressions to solve for the unknown variable 'x'. Here's a step-by-step guide:

Expanding the Products

  1. Left-hand side: The left side of the equation represents the difference of squares pattern: (a-b)(a+b) = a² - b²

    • Apply the pattern: (2x - 3)(2x + 3) = (2x)² - (3)² = 4x² - 9

  2. Right-hand side: Expand the squared term and simplify.

    • (x + 5)² = (x + 5)(x + 5) = x² + 10x + 25
    • (x + 5)² - 9 = x² + 10x + 25 - 9 = x² + 10x + 16

Solving for x

Now our equation looks like this: 4x² - 9 = x² + 10x + 16

  1. Combine like terms: Subtract x² from both sides: 3x² - 9 = 10x + 16
  2. Isolate the x term: Subtract 10x and 16 from both sides: 3x² - 10x - 25 = 0
  3. Factor the quadratic equation: Find two numbers that multiply to (3 * -25 = -75) and add up to -10. These numbers are -15 and 5.
    • (3x + 5)(x - 5) = 0
  4. Solve for x: Set each factor equal to zero and solve for x:
    • 3x + 5 = 0 => x = -5/3
    • x - 5 = 0 => x = 5

Conclusion

Therefore, the solutions to the equation (2x-3)(2x+3) = (x+5)² - 9 are x = -5/3 and x = 5.

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