Germany | A nice Math Olympiad Algebra | Can You Solve this ? | Cube Root Simplification #mathstrick

Germany | A nice Math Olympiad Algebra | Can You Solve this ? | Cube Root Simplification
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[ЛидийКлещельский-ьх]

Thank you. Bat it’s possible a little differently !
(1) (a—b)^3=a*3–b^3–3*a*b*(a—b). Therefore : (2) . Therefore (3) [16*x^2–49^2]^(1/3)=15 . Therefore (4) 16*x^2=15^3+49^2=5776 . We get Your answer.
With respect, Lidiy

[woobjun]

Letting the given,
a -b = 2 ...(eq1)
while to be
a^3 -b^3 = 98.
Cubing (eq1)
(a -b)^3 = a^3 -b^3 -3ab(a -b);
2^3 = 98 -6ab;
6ab = 90;
ab = 15 ...(eq2)
By (eq1) and (eq2),
a -15/a = 2;
a^2 -2a -15 =0;
a^2 -5a +3a -15 =0;
a(a -5) +3(a -5) =0;
(a -5)(a +3) =0;
a = 5 or -3.
Thus,
a^3 = 4x +49 = (5)^3 =125;
4x = 76; x =19
OR
a^3 = 4x +49 = (-3)^3;
4x = -76; x = -19
Therefore,
x = -19, +19

[nikitaagrawal]

I got +19, -19, +76, -76 by two different methods

[АндрейПергаев-зн]

А если возвести сразу в куб, то
После упрощений получается
16х^2-49^2=16
И никаких лишних замен нет

[qu_ark]

I found this solution easily:
Rearrange as ∛(4x+49) - 1 = ∛(4x-49) + 1. Let t equal these, so (t+1)^3 = 4x + 49 and (t-1)^3 = 4x - 49.
Adding these equations (and expanding and dividing by 2) gives t^3 + 3t = 4x (*); subtracting instead of adding gives 3t^2 + 1 = 49 (**).
The solutions to (**) are t = 4 and t = -4. Substituting these into (*) we get 4x = t(t^2 + 3) = 4*19 or -4*19, so x = 19 or -19.