Final answer:
The equation 8x³ - 125 = 0 is a difference of cubes that can be factored and solved to find that x = 2.5.
Step-by-step explanation:
To factor the equation 8x³ - 125 = 0, we can observe that both terms are perfect cubes.
The first term, 8x³, is the cube of (2x), and the second term, -125, is the cube of (-5).
This equation is actually a difference of cubes, which can be factored using the formula a³ - b³ = (a - b)(a²+ ab + b²).
Applying this formula to our equation, where a is (2x) and b is 5, we get: (2x - 5)((2x)² + (2x)(5) + (5)²)
= (2x - 5)(4x² + 10x + 25)
Now that we have factored the equation, it is equal to zero when (2x - 5) is zero or when (4x²+ 10x + 25) is zero. However, since (4x² + 10x + 25) is always positive, the only solution comes from 2x - 5 = 0.
Solving for x, we get x = 2.5.