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Question

# Find all values of c such that the domain of f(x) = x+3 / x²+3cx+6 is the set of all real numbers.

Solution

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For the domain of this function to be set of all real numbers, the denominator must never be zero:

$\begin{equation*} x^2+3cx+6\ne0 \end{equation*}$

There are no real roots of quadratic equation if the discriminant is negative.

$\begin{gather*} D\text{\textless} 0\\ b^2-4ac\text{\textless} 0\\ (3c)^2-4\cdot 1 \cdot 6 \text{\textless} 0\\ 9c^2-24 \text{\textless} 0\\ 9c^2 \text{\textless} 24\\ c^2 \text{\textless} \dfrac{24}{9}=\dfrac{8}{3}\\ c\text{\textless} \sqrt{\dfrac{8}{3}}=2\sqrt{\dfrac{2}{3}}\\ c\text{\textgreater} \sqrt{\dfrac{8}{3}}=-2\sqrt{\dfrac{2}{3}}\\ \end{gather*}$

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