Germany | A Nice Math Olympiad Algebra Problem

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[@NadiehFan]

A different approach which does not require a substitution and which does not require having to deal with fractions starts by multiplying both sides by (x − 3)² to get rid of the fraction. We can then proceed as follows x²(x − 3)² + 9x² = 16(x − 3)² x²(x² − 6x + 9) + 9x² = 16(x − 3)² x²(x² − 6x + 18) = 16(x − 3)² (x² − 3(x − 3) + 3(x − 3))(x² − 3(x − 3) − 3(x − 3)) = 16(x − 3)² (x² − 3x + 9)² − 9(x − 3)² = 16(x − 3)² (x² − 3x + 9)² − 25(x − 3)² = 0 (x² − 3x + 9 + 5(x − 3))(x² − 3x + 9 − 5(x − 3)) = 0 (x² + 2x − 6)(x² − 8x + 24) = 0 x = −1 + √7 ⋁ x = −1 − √7 ⋁ x = 4 + 2i√2 ⋁ x = 4 − 2i√2

[@laurentbeyer2458]

I would have done same 👍🏻

[@TapanDas-ei3lw]

You're also a great genius ❤❤❤

[@michaelhuppertz6738]

Nice solution specially how got the (x-3) out of 6x^3+18x^2, neat trick.

[@ajmiller7102]

Excellent. Straight to the solutions. Well done!

[@michellewelchman9245]

I enjoyed your work and your explanation. Excellent and well done! These problems can be long and it took nearly three pages for this one! Lol Sometimes it feels like you're going down a math rabbit hole with no end but in the end when you do reach a solution (s) it's all good.

[@peterbyrne6394]

Thank you. very nice solved equation.

[@NadiehFan]

Yet another method is to note that (x − 3)² = x² − 6x + 9 and therefore x² = (x − 3)² + 6x − 9 so we can rewrite the equation x² + (3x/(x − 3))² = 16 as (x − 3)² + 6x − 9 + (3x/(x − 3))² = 16 or (x − 3)² + 6x + (3x/(x − 3))² − 25 = 0 Here (x − 3)² + 6x + (3x/(x − 3))² is a perfect square because 6x is twice the product of (x − 3) and 3x/(x − 3) and we also have 25 = 5² so we get ((x − 3) + 3x/(x − 3))² − 5² = 0 which gives us a difference of two squares at the left hand side whereas the right hand side is zero. This allows us to factor the left hand side using the difference of two squares identity and then apply the zero product property. However, before doing so we first multiply both sides by (x − 3)² to eliminate the fraction so we get ((x − 3)² + 3x)² − (5(x − 3))² = 0 (x² − 3x + 9)² − (5x − 15)² = 0 (x² + 2x − 6)(x² − 8x + 24) = 0 x = −1 + √7 ⋁ x = −1 − √7 ⋁ x = 4 + 2i√2 ⋁ x = 4 − 2i√2

[@TapanDas-ei3lw]

Never ending marathon mathematics and brain 🧠 puzzled stories ever. Einstein newton and all superb mathematicians brain to be needed to understood not like me the stupid one..not getting any sweet juice extract from your mathematics, all energies i have lost to get it understood sir .really sir you've the genius brain and you're great.❤❤❤after this life i shall pray to God ,make me genius like you.❤❤❤❤❤

[@ahadjonxamidov8940]

Zo‘r

[@user-cc4zk1ec8c]

It can be solved by converting x to a trigonometric function. (x/4)^2 + (3x/(4(x-3)))^2 = 1 x/4 = cos 3x/(4(x-3)) = sin 16(cs)^2 - 18cs - 9 = 0

[@mahmoudibra5822]

Beautiful

[@learncommunolizer]

Thank you very much

[@user-ix3lv9qz5b]

It's a troubled solution.

[@tamarshahverdyan2723]

# 30 #

[@martinreichard3243]

I prefer to take the square root first and then expand with x-3. This is faster.

[@oxfordmathseries]

I prefer to get rid of fraction first, then solution will be of 2 minutes only

[@Ron_DeForest]

Would have been way faster and easier. So much less time to multiply through then gather terms.

[@user-vo4qp5bm7p]

Почему в ответе Вы сократили на 2..???? Так нельзя!!!

[@tatyanasnisarenko1627]

Х=2

[@mohamedsalah5525]

4-

[@spelleg11]

Great.

[@samswift4921]

C 1.4 hrs or 84 minutes.

[@TheTeller2012]

X-x/3-1=0

[@rajanisarmah6307]

Aare vaai itni si sota math aur kitna lomba korega??

[@bandoubenamar2802]

I no 😂

[@divineinspiration3342]

Spreco di carta

[@anuarsalomao9748]

COMPLICADO

[@anuarsalomao9748]

MAS É ORGANIZADO

[@jamielunes1841]

I want to hear a German 😂

[@Limited_Light]

Warum sagen sie das?

[@himadrikhanra7463]

1/10..?