Finding value of k for which fg(x)=k has equal roots?
I've been going through this community and I Find this really helpful. About me(I know I should be precise but ya), I'm just a highschool student who can't afford any coachings/schools. Self schooling being my only option I'm trying to teach myself mathematics from some torrented books. I am on functions and their graphs and stuck with one question
The functions f and g are defined for x ∈ R by f(x) =4x − 2x^2; g(x)= 5x + 3.
(i) Find the range of f. (ii) Find the value of the constant k for which the equation gf(x) = k has equal roots.
Now, I do understand the composition of functions but I just don't understand what they are asking in this case, g(fx()=k would result in
20x-10x^2 +3=k
Now this is a quadratic equation of second degree which should have 2 roots/solutions, but that's the case if the right hand side was zero and not k. I have absolutely no idea how to tackle this question and what's being asked in part ii of the question. Anyhelp would be highly appreciated
$\endgroup$ 21 Answer
$\begingroup$Your equation could be rewritten as $$10x^2-20x+(k-3)=0.$$
Recall that the roots of a quadratic equation $ax^2+bx+c=0$ are given by $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ So the two roots are equal when $b^2-4ac=0$. Can you apply this to your problem?
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