(x%5E2-64)%3D0.jpeg "(x^3+125)(x^2-64)=0")
Solving the Equation (x³ + 125)(x² - 64) = 0
This equation involves a product of two expressions equaling zero. The key to solving this type of equation is using the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
Let's break down the equation and apply the Zero Product Property:
1. Identify the factors:
The equation is already factored: (x³ + 125)(x² - 64) = 0
Our two factors are:
- x³ + 125
- x² - 64
2. Set each factor equal to zero:
We need to find the values of 'x' that make each factor equal to zero.
- x³ + 125 = 0
- x² - 64 = 0
3. Solve for 'x' in each equation:
-
x³ + 125 = 0
- Subtract 125 from both sides: x³ = -125
- Take the cube root of both sides: x = -5
-
x² - 64 = 0
- Add 64 to both sides: x² = 64
- Take the square root of both sides: x = ±8
4. Solutions:
Therefore, the solutions to the equation (x³ + 125)(x² - 64) = 0 are:
- x = -5
- x = 8
- x = -8
In conclusion: By applying the Zero Product Property, we successfully found the three solutions to the given equation. These solutions represent the values of 'x' that make the entire equation true.