Problem 8 · The exponential equation
Common factor $19^{t-1}$: the exponent must vanish.
Integer answer, at most 4 digitsMultiply the solutions of this equation. If the result is negative, give the answer in absolute value.
$$19^{x^2+4x-21} - 19^{x^2+4x-22} = \frac{54}{57}$$
Copa Cangur · SCM
Hard
Closed answer
Reasoned solution
Key idea: simplify the fraction and factor out the smaller power.
On one hand $\tfrac{54}{57} = \tfrac{18}{19}$. On the other, with $t = x^2 + 4x - 21$:
$$19^{t} - 19^{t-1} = 19^{t-1}(19 - 1) = 18 \cdot 19^{t-1}.$$
$$18 \cdot 19^{t-1} = \frac{18}{19} \;\Longrightarrow\; 19^{t-1} = 19^{-1} \;\Longrightarrow\; t = 0.$$
Hence $x^2 + 4x - 21 = 0$, that is $(x+7)(x-3) = 0$: solutions $x = -7$ and $x = 3$. Their product is $-21$, in absolute value $21$. (By Vieta, the product is just the constant term, $-21$.)
Answer: 21