Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 1480. (April 2018)

C. 1480. Solve the equation

\(\displaystyle \frac{x^3-7x+6}{x-2}=\frac{2x+14}{x+2} \)

on the set of integers.

(5 pont)

Deadline expired on May 10, 2018.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. A nevezők miatt \(\displaystyle x≠2\) és \(\displaystyle x≠-2\). Alakítsuk szorzattá a harmadfokú kifejezést:

\(\displaystyle x^3-7x+6=x^3-2x^2+2x^2-4x-3x+6=(x-2)(x^2+2x-3)= (x-2)(x+3)(x-1).\)

Így az egyenletünk

\(\displaystyle \frac{(x-2)(x+3)(x-1)}{x-2}=\frac{2x+4+10}{x+2}.\)

Bal oldalon \(\displaystyle (x-2)\neq=0\)-val egyszerűsítve, jobb oldalon \(\displaystyle (x+2)\neq=0\)-val tagonként leosztva:

\(\displaystyle (x+3)(x-1)=2+\frac{10}{x+2}.\)

A bal oldal egész, így a jobb oldal, és ezért \(\displaystyle \frac{10}{x+2}\) is az. Tehát \(\displaystyle z=x+2\) lehetséges értékei \(\displaystyle 10\) osztói: \(\displaystyle -10\), \(\displaystyle -5\), \(\displaystyle -2\), \(\displaystyle -1\), \(\displaystyle 1\), \(\displaystyle 2\), \(\displaystyle 5\), \(\displaystyle 10\).

\(\displaystyle (z+1)(z-3)=2+\frac{10}{z}\)

\(\displaystyle z\) \(\displaystyle -10\) \(\displaystyle -5\) \(\displaystyle -2\) \(\displaystyle -1\) \(\displaystyle 1\) \(\displaystyle 2\) \(\displaystyle 5\) \(\displaystyle 10\)
bal oldal \(\displaystyle 117\) \(\displaystyle 32\) \(\displaystyle 5\) \(\displaystyle 0\) \(\displaystyle -4\) \(\displaystyle -3\) \(\displaystyle 12\) \(\displaystyle 77\)
jobb oldal \(\displaystyle 1\) \(\displaystyle 0\) \(\displaystyle -3\) \(\displaystyle -8\) \(\displaystyle 12\) \(\displaystyle 7\) \(\displaystyle 4\) \(\displaystyle 3\)

Az egyenletnek nincs megoldása az egész számok halmazán.


Statistics:

134 students sent a solution.
5 points:59 students.
4 points:22 students.
3 points:20 students.
2 points:8 students.
1 point:13 students.
0 point:10 students.
Unfair, not evaluated:2 solutionss.

Problems in Mathematics of KöMaL, April 2018