College

Solve for all complex roots of [tex]5x^5 - 45x^3 + 70x = 0[/tex]. If necessary, leave your answer as a radical or with [tex]i[/tex].

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:

  1. x = 0
  2. 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:

  1. z = 2
  2. z = 7

Substituting x^2 back in for z, we get:

  1. x^2 = 2
  2. x^2 = 7

By taking the square root of both sides, we find:

  1. x = ±√2
  2. x = ±√7

So the complex roots of the equation are x = 0, ±√2, ±√7.


Learn more about Complex Roots of Equations here:

https://brainly.com/question/12935822


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