(x%2B8)%3D42.jpeg "(x-3)(x+8)=42")
Solving the Quadratic Equation: (x-3)(x+8) = 42
This article will guide you through the process of solving the quadratic equation (x-3)(x+8) = 42. We'll break it down into clear steps, ensuring you understand each part of the solution.
1. Expanding the Equation
The first step is to expand the left side of the equation by multiplying the binomials:
(x - 3)(x + 8) = 42 x² + 5x - 24 = 42
2. Rearranging into Standard Quadratic Form
To solve a quadratic equation, it's crucial to have it in standard form:
ax² + bx + c = 0
Therefore, we need to move the constant term (42) to the left side:
x² + 5x - 24 - 42 = 0 x² + 5x - 66 = 0
3. Factoring the Quadratic Expression
Now, we need to factor the quadratic expression on the left side. We are looking for two numbers that:
- Multiply to -66 (the constant term)
- Add up to 5 (the coefficient of the x term)
The numbers -6 and 11 satisfy these conditions:
(-6) * 11 = -66 -6 + 11 = 5
Therefore, we can factor the expression as:
(x - 6)(x + 11) = 0
4. Solving for x
For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve:
- x - 6 = 0 => x = 6
- x + 11 = 0 => x = -11
Conclusion
The solutions to the quadratic equation (x-3)(x+8) = 42 are x = 6 and x = -11.
This method demonstrates how to solve quadratic equations by factoring. Remember that factoring is a powerful technique for simplifying expressions and finding solutions to polynomial equations.