Find the Domain fourth root of x^2+3x

Problem

√(4,x2+3*x)

Solution

1. Identify the condition for the domain of an even root. For the fourth root to be defined as a real number, the radicand must be greater than or equal to zero.

x2+3*x≥0

2. Factor the quadratic expression to find the critical points where the expression equals zero.

x*(x+3)≥0

3. Determine the critical points by setting each factor to zero.

x=0

x+3=0⇒x=−3

4. Test the intervals created by the critical points x=−3 and x=0 to see where the inequality holds true. The intervals are (−∞,−3] [−3,0] and [0,∞)

5. Evaluate a test point in each interval. For x=−4 (−4)*(−1)=4≥0 (True). For x=−1 (−1)*(2)=−2≥0 (False). For x=1 (1)*(4)=4≥0 (True).

6. Combine the intervals where the inequality is satisfied to express the domain.

(−∞,−3]∪[0,∞)

Final Answer

Domain of √(4,x2+3*x)=(−∞,−3]∪[0,∞)

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