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(x+5)(x-5)+x(x-5)=0

2 min read Jun 17, 2024
(x+5)(x-5)+x(x-5)=0

Solving the Equation: (x+5)(x-5) + x(x-5) = 0

This equation represents a quadratic equation that can be solved using a few different methods. Here's a breakdown of the steps:

1. Factoring

  • Expand the equation:

    • (x+5)(x-5) = x² - 25
    • x(x-5) = x² - 5x
    • Therefore, the equation becomes: x² - 25 + x² - 5x = 0

  • Combine like terms: 2x² - 5x - 25 = 0

  • Factor the quadratic: (2x+5)(x-5) = 0

  • Set each factor equal to zero and solve for x:

    • 2x + 5 = 0 => x = -5/2
    • x - 5 = 0 => x = 5

2. Using the Quadratic Formula

The quadratic formula provides a solution for any equation of the form ax² + bx + c = 0.

  • Identify the coefficients:

    • a = 2
    • b = -5
    • c = -25

  • Plug the values into the quadratic formula:

    • x = (-b ± √(b² - 4ac)) / 2a
    • x = (5 ± √((-5)² - 4 * 2 * -25)) / 2 * 2
    • x = (5 ± √(225)) / 4
    • x = (5 ± 15) / 4

  • Solve for the two possible values of x:

    • x = (5 + 15) / 4 = 5
    • x = (5 - 15) / 4 = -5/2

Solution

Therefore, the solutions to the equation (x+5)(x-5) + x(x-5) = 0 are x = 5 and x = -5/2.

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