# Consider the two functions $f(x)=x^2+2bx+9; \; g(x)= 3a(x+b)$. Find area of the coordinates for non-intersecting graph

Consider the two functions $$f(x)=x^2+2bx+9; \; g(x)= 3a(x+b)$$, where $$a,b$$ are real numbers.

Each pair of $$(a,b)$$ may be considered to be coordinates of a point in the x-y plane. Let $$S$$ be the set of all such points $$(a,b)$$ for which the graph of $$y=f(x)$$ and $$y=g(x)$$ do not intersect.

Find the area of the region defined by S.

I understood what the question meant but I am unable to find an elegant and profound way of solving this question. Someone told me to equate $$f(x)=g(x)$$ and then find the value of $$x$$. I am not understanding why we have to equate these two. Can you explain this with an example and give a hint of starting the question?

Hint: solve them simultaneously and apply the condition that the discriminant of this quadratic must be negative. This leads to a simple inequality in $$a$$ and $$b$$ which defines the area inside a well-known shape.