Find the Domain (2x^2+16x+30)/(5x^2+13x-6)

Problem

(2*x2+16*x+30)/(5*x2+13*x−6)

Solution

1. Identify the condition for the domain of a rational function, which requires that the denominator cannot be equal to zero.

5*x2+13*x−6≠0

2. Factor the quadratic expression in the denominator using the grouping method or the quadratic formula.

5*x2+15*x−2*x−6≠0

5*x*(x+3)−2*(x+3)≠0

(5*x−2)*(x+3)≠0

3. Solve for the excluded values by setting each factor to zero.

5*x−2=0⇒x=2/5

x+3=0⇒x=−3

4. State the domain by excluding these specific values from the set of all real numbers.

Domain={[x∈ℝ,x≠−3,x≠2/5]}

Final Answer

Domain=(−∞,−3)∪(−3,2/5)∪(2/5,∞)

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