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

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

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

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

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.