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(x-3)(x-2)=(x+5)(2x-3)+21

2 min read Jun 17, 2024
(x-3)(x-2)=(x+5)(2x-3)+21

Solving the Equation: (x-3)(x-2) = (x+5)(2x-3) + 21

This article will guide you through the process of solving the equation: (x-3)(x-2) = (x+5)(2x-3) + 21.

Expanding and Simplifying

First, we need to expand both sides of the equation by multiplying the terms:

  • Left Side:
    • (x-3)(x-2) = x² - 2x - 3x + 6 = x² - 5x + 6
  • Right Side:
    • (x+5)(2x-3) + 21 = 2x² - 3x + 10x - 15 + 21 = 2x² + 7x + 6

Now the equation looks like this: x² - 5x + 6 = 2x² + 7x + 6

Rearranging the Equation

To solve for x, we need to get all the terms on one side of the equation:

  • Subtract x² from both sides: -5x + 6 = x² + 7x + 6
  • Subtract 6 from both sides: -5x = x² + 7x
  • Subtract 7x from both sides: -12x = x²

Now the equation is: x² + 12x = 0

Solving for x

This equation is a quadratic equation. We can solve for x by factoring:

  • Factor out x: x(x + 12) = 0

For this equation to be true, either x = 0 or x + 12 = 0. Therefore, the solutions are:

  • x = 0
  • x = -12

Conclusion

Therefore, the solutions to the equation (x-3)(x-2) = (x+5)(2x-3) + 21 are x = 0 and x = -12.

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