I am given:
$f(x) = 4x-2k\ $ and $\ g(x) = \dfrac{9}{2-x}$ ($x \not = 2$)
The question is:
Find the values for k for which the equation $fg(x) = x$ has two equal roots.
So $f(g(x))=\dfrac{4\cdot (9)}{2-x} -2k = \dfrac{36}{2-x} -2k$
$k$ is given as a constant (this means it is a number, and does not include variables?)
In the past I have used the discriminant of the quadratic formula to solve if something has equal roots. I'm not too sure what to do with this equation.
Best Answer
$f(g(x))=\dfrac{36}{2-x} -2k$
$f(g(x))=x\rightarrow \dfrac{36}{2-x} -2k=x \rightarrow x^2 - 2x(1 - k) - 4(k - 9) = 0$
Solution for $x$ are equal if and only if discriminant $\Delta=0$
$4(1-k)^2+16(k-9)=0$
$k^2+2k-35=0\rightarrow k_1=-7;\;k_2=5$