(x+4)(x^2-4x+16)=0

Jasonbradley

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Jun 16, 2024

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Solving the Equation (x+4)(x^2-4x+16) = 0

This equation is already factored, making it relatively straightforward to solve. Here's how we can approach it:

Understanding the Equation

The equation represents the product of two factors:

• (x+4): This is a linear factor.
• (x^2-4x+16): This is a quadratic factor.

For the product of these factors to equal zero, at least one of the factors must be equal to zero.

Solving for x

Let's apply the zero product property:

1. Set each factor equal to zero:

   • x + 4 = 0
   • x^2 - 4x + 16 = 0

2. Solve the linear equation:

   • x = -4

3. Solve the quadratic equation:

   • This quadratic equation doesn't factor easily, so we can use the quadratic formula:

     • x = [-b ± √(b^2 - 4ac)] / 2a
     • Where a = 1, b = -4, and c = 16

   • Substituting the values:

     • x = [4 ± √((-4)^2 - 4 * 1 * 16)] / 2 * 1
     • x = [4 ± √(-48)] / 2
     • x = [4 ± 4√(-3)] / 2
     • x = 2 ± 2√(-3)
     • x = 2 ± 2i√3 (where 'i' is the imaginary unit, √-1)

Solutions

Therefore, the solutions to the equation (x+4)(x^2-4x+16) = 0 are:

• x = -4
• x = 2 + 2i√3
• x = 2 - 2i√3

The equation has one real solution (x = -4) and two complex solutions (x = 2 + 2i√3 and x = 2 - 2i√3).