How to solve this nice math Exponential algebra problem | Olympiad Question | a=?,b=?

#exponents #maths #olympiad #algebra #integral
math olympiad algebra problem,nice algebra problem,math olympiad questions,math olympiad problems,math olympiad algebra,math olympiad question,math olympiad preparation,math olympiad,math olympiad problem,algebra problem,math problems,algebra problems,learn how to solve exponential equation quickly,olympiad math problems,olympiad mathematics,algebra,a nice algebra problem,algebra math olympiad,viral math olympiad problem | how to solve

[guyhoghton]

_a² - b² = 28, ab = 48_
Set _(t - a²)(t + b²) = 0_ ... ①
⇒ _t² - (a² - b²)t - (ab)² = 0_
⇒ _t² - 28t - 48² = 0_
⇒ _t² - 28t - (8²)(6²) = 0_
⇒ _(t - 8²)(t + 6²) = 0_
⇒ _(a², b²) = (8², 6²)_ or _(-6², -8²)_ by comparison with ①
⇒ _(a, b) = (±8, ±6)_ or _(±6i, ±8i)_
But _ab = 48_ ∴ _(8, -6), (-8, 6), (6i, 8i)_ and _(-6i, -8i)_ are extraneous solutions.
∴ *_(a, b) = (8, 6), (-8, -6), (6i, -8i)_** or **_(-6i, 8i)_*

[RealQinnMalloryu]

7^4 7^1 ➖ 4 1^1 ➖ 2^2 1^2 (b ➖ 2a+1). 2^24 2^12 2^6 2^3^2 1^1^2 1^2 (ab ➖ 2ab+1).