Answer :
Final answer:
The complex roots of the equation 5x^5 - 45x^3 + 70x = 0 are 0, ±√2, ±√7.
Explanation:
Solving the Equation 5x^5 - 45x^3 + 70x = 0
To solve the equation, we can first factor out a common factor of x:
5x(x^4 - 9x^2 + 14) = 0
Now we have two factors:
- x = 0
- x^4 - 9x^2 + 14 = 0
For the second factor, let's substitute a variable to simplify the equation. Let z = x^2:
z^2 - 9z + 14 = 0
Now we have a quadratic equation which we can solve by factoring:
(z - 2)(z - 7) = 0
This gives us two possible values for z:
- z = 2
- z = 7
Substituting x^2 back in for z, we get:
- x^2 = 2
- x^2 = 7
By taking the square root of both sides, we find:
- x = ±√2
- x = ±√7
So the complex roots of the equation are x = 0, ±√2, ±√7.
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