Question

Determine the roots of the function f(x) = (5^2x −6)^2 −(5^2x −6)−12.

Answer 1

Let's note first $t:=5^{2x}-6$ , then we can write:

$t^2-t-12=0$

It is an equation of second order so we can the solutions pretty easily (either with Vieta's formulas, either with the explicit formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ ):

$t_1=4,t_2=-3$

Now let's come back from our substitution :

$5^{2x_1}-6=4, 5^{2x_2}-6=-3$

Therefore we find two solutions :

$x_1=\frac{1}{2}\log_5(10), x_2=\frac{1}{2}\log_5(3)$