domain of sqrt{-x^3+5x^2+9x-45}

Question

domain of sqrt{-x^3+5x^2+9x-45}

Answer 1

Assuming that x is restricted to the set of real numbers, we can't take square roots of negative numbers.  So, -x³+5x²+9x-45 ≥0.

 

Factor by grouping:  -x²(x-5) + 9(x-5) ≥ 0

 

                             (x-5)(-x²+9) ≥ 0

 

                             (x-5)(-x+3)(x+3) ≥ 0

 

The left side is equal to zero when x = 5, 3, or -3.

 

_________l___________l____________l__________

              -3                 3                   5

 

Choose a test point from each of the intervals (-∞,-3), (-3,3), (3,5), (5,∞) and plug into the factored form above.  

 

Interval          Test Point        Sign of (x-5)(-x+3)(x+3)

 

(-∞, -3)               -4                    positive

(-3,3)                   0                    negative

(3,5)                    4                    positive

(5,∞)                   6                    negative

 

Since we are trying to solve the inequality (x-5)(-x+3)(x+3)≥0, we choose the intervals in the table above for which we got "positive" in the rightmost column. We also include the endpoints (not -∞) as part of the solution set since, at those points, (x-5)(-x+3)(x+3) = 0.

  

Solution set:  (-∞, -3] ∪ [3,5]

  

Answer 2

Solve the inequality radicand ≥ 0 to find values of x that must be excluded since taking the square root of a negative must be disallowed.  Note: I think that you can GROUP FACTOR the radicand.