(x+8)(x-7)(x2-2x+5)=0

Jasonbradley

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Jun 17, 2024

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Solving the Cubic Equation: (x+8)(x-7)(x^2-2x+5) = 0

This equation represents a cubic function, meaning it has a highest power of 3. To solve for the values of x that satisfy this equation, we can utilize the zero product property.

The Zero Product Property

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

In our equation, we have three factors:

1. (x+8)
2. (x-7)
3. (x^2-2x+5)

Therefore, for the entire equation to equal zero, at least one of these factors must equal zero.

Solving for x

1. (x+8) = 0

   • Subtracting 8 from both sides, we get x = -8

2. (x-7) = 0

   • Adding 7 to both sides, we get x = 7

3. (x^2-2x+5) = 0

   • This quadratic equation doesn't factor easily. We can use the quadratic formula to solve for x:
     • x = (-b ± √(b^2 - 4ac)) / 2a
     • Where a = 1, b = -2, and c = 5 (from the quadratic equation)
     • Substituting the values, we get:
       • x = (2 ± √((-2)^2 - 4 * 1 * 5)) / (2 * 1)
       • x = (2 ± √(-16)) / 2
       • x = (2 ± 4i) / 2 (where 'i' is the imaginary unit, √-1)
       • x = 1 ± 2i

Solutions

Therefore, the solutions for the equation (x+8)(x-7)(x^2-2x+5) = 0 are:

• x = -8
• x = 7
• x = 1 + 2i
• x = 1 - 2i

The equation has two real solutions (x = -8 and x = 7) and two complex solutions (x = 1 + 2i and x = 1 - 2i).