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

Germany | A nice Math Olympiad Algebra | Can You Solve this ? | Cube Root Simplification
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[@ЛидийКлещельский-ь3х]

Thank you. Bat it’s possible a little differently ! (1) (a-b)^3=a*3-b^3-3*a*b*(a-b). Therefore : (2) 2^3=(4*x+49)-(4*x-49)-3*(16*x^2-49^2)^(1/3)*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

[@woobjun2582]

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

[@nikitaagrawal6215]

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

[@superacademy247]

Thanks for your support. Please 🙏 share the two methods

[@АндрейПергаев-з4н]

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

[@l.w.paradis2108]

This is great.

[@superacademy247]

Thanks 👍💯 for your support

[@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.