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(x+4)(x+5)-30=0

3 min read Jun 16, 2024
(x+4)(x+5)-30=0

Solving the Equation (x+4)(x+5) - 30 = 0

This article will guide you through the steps of solving the quadratic equation (x+4)(x+5) - 30 = 0.

Expanding and Simplifying the Equation

  1. Expand the product: Begin by expanding the product of the binomials: (x+4)(x+5) = x² + 5x + 4x + 20

  2. Combine like terms: Simplify the expanded expression: x² + 5x + 4x + 20 = x² + 9x + 20

  3. Rewrite the equation: Now we have: x² + 9x + 20 - 30 = 0

  4. Simplify: Combine the constant terms: x² + 9x - 10 = 0

Solving the Quadratic Equation

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

1. Factoring:

  • Find two numbers: Find two numbers that add up to 9 (the coefficient of the x term) and multiply to -10 (the constant term). These numbers are 10 and -1.
  • Factor the equation: Rewrite the equation using these numbers: (x + 10)(x - 1) = 0
  • Solve for x: For the product of two factors to be zero, at least one of them must be zero. Therefore: x + 10 = 0 or x - 1 = 0 x = -10 or x = 1

2. Quadratic Formula:

  • Identify coefficients: In the equation x² + 9x - 10 = 0, a = 1, b = 9, and c = -10.
  • Apply the formula: The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a
  • Substitute values: x = (-9 ± √(9² - 4 * 1 * -10)) / (2 * 1) x = (-9 ± √(121)) / 2 x = (-9 ± 11) / 2
  • Solve for x: x = (-9 + 11) / 2 = 1 x = (-9 - 11) / 2 = -10

Conclusion

We have successfully solved the equation (x+4)(x+5) - 30 = 0 using both factoring and the quadratic formula. The solutions are x = 1 and x = -10.

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