A Nice Math Olympiad Algebra Problem | Find a and b?

[@НеллиПшено]

Решаем методом устного счета.
a(a-b)=7х2
a=7
b=7-2=5
ab=7x5=35
Good luck!

[@SALogics]

Very nice! ❤

[@key_board_x]

a² - ab = 14
a.(a - b) = 14
(a - b) = 14/a ← equation (1)
b² + ab = 60
b.(a + b) = 60
(a + b) = 60/b ← equation (2)
a² - ab = 14
b² + ab = 60
-------------------------------------------sum
a² + b² = 74
b² = 74 - a² ← equation (3)
a² - b² = (a + b).(a - b) → recall (1) & (2)
a² - b² = (60/b).(14/a)
a² - b² = 840/(ab)
a² - (74 - a²) = 840/(ab)
a² - 74 + a² = 840/(ab)
2a² - 74 = 840/(ab)
a² - 37 = 420/(ab) → given: a² - ab = 14 → ab = a² - 14
a² - 37 = 420/(a² - 14)
(a² - 37).(a² - 14) = 420
a⁴ - 14a² - 37a² + 518 = 420
a⁴ - 51a² + 98 = 0
Δ = (- 51)² - (4 * 98) = 2601 - 392 = 2209 = 47²
a² = (51 ± 47)/2
First case: a² = (51 + 47)/2 = 49 → a = ± 7
First possibility: a = 7 → recall: ab = a² - 14 → 7b = 49 - 14 → b = 5
Second possibility: a = - 7 → recall: ab = a² - 14 → - 7b = 49 - 14 → b = - 5
Second case: a² = (51 - 47)/2 = 4 → a = ± 2
First possibility: a = 2 → recall: ab = a² - 14 → 2b = 16 - 14 → b = 1
Second possibility: a = - 2 → recall: ab = a² - 14 → - 2b = 16 - 14 → b = - 1
Summarize (a ; b)
(7 ; 5)
(- 7 ; - 5)
(2 ; 1)
(- 2 ; - 1)

[@SALogics]

Very nice! ❤

[@cabasantbab]

7 &5

[@SALogics]

There are 3 other solutions! ❤