Solving the Equation (x+3)(x8) = 60
This article will guide you through solving the equation (x+3)(x8) = 60. We'll explore the steps involved in finding the solutions for x.
1. Expanding the Equation
First, we need to expand the left side of the equation by multiplying the binomials:
(x+3)(x8) = x²  8x + 3x  24
Simplifying the equation, we get:
x²  5x  24 = 60
2. Transforming into a Quadratic Equation
Next, we need to move all terms to one side of the equation to get a standard quadratic equation:
x²  5x  24  60 = 0
This simplifies to:
x²  5x  84 = 0
3. Factoring the Quadratic Equation
Now, we can factor the quadratic equation to find the solutions for x. We need to find two numbers that add up to 5 (the coefficient of the x term) and multiply to 84 (the constant term).
The numbers 12 and 7 satisfy these conditions:
(12) + 7 = 5 (12) * 7 = 84
Therefore, we can factor the quadratic equation as:
(x  12)(x + 7) = 0
4. Solving for x
For the product of two terms to be zero, at least one of the terms must be zero. So, we have two possible solutions:

x  12 = 0 This gives us x = 12

x + 7 = 0 This gives us x = 7
Conclusion
The solutions to the equation (x+3)(x8) = 60 are x = 12 and x = 7.