(6%2B2%2F3n)(n-2)%3D0.jpeg "(2-n)(6+2/3n)(n-2)=0")
Solving the Equation: (2-n)(6+2/3n)(n-2)=0
This equation is a cubic equation, meaning the highest power of 'n' is 3. To solve it, we can use the following steps:
1. Factor the Equation:
The equation is already factored, which makes our lives easier! We have three factors:
- (2-n)
- (6+2/3n)
- (n-2)
2. Set Each Factor to Zero:
Since the product of these factors equals zero, at least one of them must be equal to zero. So, we set each factor equal to zero and solve for 'n':
- 2 - n = 0
- n = 2
- 6 + 2/3n = 0
- 2/3n = -6
- n = -9
- n - 2 = 0
- n = 2
3. The Solutions:
We have found three solutions to the equation:
- n = 2 (This solution appears twice)
- n = -9
4. Checking the Solutions:
We can substitute each value of 'n' back into the original equation to verify our solutions.
- For n = 2:
- (2-2)(6+2/3*2)(2-2) = 0 * (6+4/3) * 0 = 0
- For n = -9:
- (2+9)(6-2/3*9)(-9-2) = 11 * 0 * (-11) = 0
Conclusion:
The solutions to the equation (2-n)(6+2/3n)(n-2)=0 are n = 2 (with multiplicity of 2) and n = -9.