(x^2+4)(x+4)(x-8) =0

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x² + 4)(x + 4)(x - 8) = 0

This equation presents a unique opportunity to use the Zero Product Property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

Let's break down the equation:

• (x² + 4): This factor represents a quadratic expression.
• (x + 4): This factor is a simple linear expression.
• (x - 8): This factor is also a simple linear expression.

To solve the equation, we need to find the values of 'x' that make each of these factors equal to zero.

Solving for the Factors:

1. (x² + 4) = 0:

   • Subtracting 4 from both sides: x² = -4
   • Taking the square root of both sides: x = ±√(-4)
   • Since the square root of a negative number is imaginary, we have two solutions: x = 2i and x = -2i.

2. (x + 4) = 0:

   • Subtracting 4 from both sides: x = -4

3. (x - 8) = 0:

   • Adding 8 to both sides: x = 8

Solutions:

Therefore, the solutions to the equation (x² + 4)(x + 4)(x - 8) = 0 are:

• x = 2i
• x = -2i
• x = -4
• x = 8

These solutions represent the values of 'x' that make the entire equation true because they make one or more of the factors equal to zero.