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

2 min read Jun 17, 2024
(x-3)(2x+1)=x(x+5)

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

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

Expanding and Simplifying

  1. Expand the brackets:

    • On the left side, we use the distributive property (or FOIL method): (x-3)(2x+1) = 2x² + x - 6x - 3 = 2x² - 5x - 3
    • On the right side: x(x+5) = x² + 5x

  2. Rearrange the equation:

    • Move all terms to one side to get a quadratic equation: 2x² - 5x - 3 - x² - 5x = 0

  3. Combine like terms: x² - 10x - 3 = 0

Solving the Quadratic Equation

Now we have a quadratic equation in the standard form: ax² + bx + c = 0. There are several methods to solve this:

  • Factoring: Try to find two numbers that multiply to -3 and add up to -10. In this case, it's not easy to factor directly.

  • Quadratic Formula: The most reliable method for solving quadratic equations: x = (-b ± √(b² - 4ac)) / 2a

    In our equation: a = 1, b = -10, and c = -3 x = (10 ± √((-10)² - 4 * 1 * -3)) / (2 * 1) x = (10 ± √(112)) / 2 x = (10 ± 4√7) / 2 x = 5 ± 2√7

Conclusion

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

  • x = 5 + 2√7
  • x = 5 - 2√7

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