A Nice Math Olympiad Problem | You should know this trick!

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8 ай бұрын

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How solve ³√(X+49) - ³√(X-49) = 2
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Пікірлер: 27

@kurtecaranum3047 8 ай бұрын

Why not just cube the whole thing?? Using your work, (a-b)³ = 2³ a³ - 3ab(a-b) - b³ = 8 98 - 3*ab*2 = 8 ab = 15 (ab)³ = (x+49)(x-49) = 15³

@paul.njihia 8 ай бұрын

Why not substitute ab=15 into the original equations a=3√(x+49) and b=3√(x-49) where a×b=3√(x^2-49^2). The subsequent equation will be x^2=15^3+49^2=5776, x=+76 or -76.

@percy9228 8 ай бұрын

nice

@schlingel0017 5 ай бұрын

Harder to calculate. I need a calculator to find out square of 49, cubic of 15 and square root of 5776

@mathenligne5657 8 ай бұрын

Nice job thanks 🎉🎉🎉

@pasqualguidajunior5065 7 ай бұрын

A really difficult exercise to solve. Congratulations on the solution!

@learncommunolizer 7 ай бұрын

Thank you very much!!

@Phanatomicool 8 ай бұрын

This is the way I solved the problem (I don’t know how to do math notation stuffs, so for this sqrt() is the square root function, crt() is cube root function, a^b means “a to the b power”, - is subtraction, + is addition, / is division and * is multiplication, +pm- means “plus or minus”) Solve for x: crt(x+49)-crt(x-49)=2 Let a = crt(x+49) Let b = crt(x-49) a-b=2 a=2+b a^3 = x + 49 b^3 = x - 49 Isolate variable x and substitute a for b: a^3 + b^3 = x+49+x-49 = 2x (a^3 + b^3)/2 = x (Because a=2+b): ((2+b)^3 + b^3)/2 = x (2+b)^3+b^3 = 2x Finding the polynomial representation of “(2+b)^3”: (2+b)^3 = (2+b)(2+b)(2+b) = (b^2+4b+4)(2+b) = 2b^2+8b+8+b^3+4b^2+4b = b^3+6b^2+12b+8 Creating a quadratic equation: (2+b)^3+b^3 = 2x (2+b)^3 = 2x-b^3 (2+b)^3 = 2x-(x-49) (Because crt(x-49)^3 = x-49) b^3 + 6b^2 + 12b + 8 = 2x - x + 49 b^3 + 6b^2 + 12b + 8 = x+49 b^3 + 6b^2 + 12b + 8 = a^3 6b^2+12b+8 = a^3-b^3 6b^2+12b+8 = x+49 - (x-49) = x-x+49+49 = 98 6b^2+12b+8 = 98 6b^2+12b-90=0 Solving the quadratic equation: 6b^2+12b-90=0 (Quadratic Formula): b = (-12 +pm- sqrt(12^2-4(6)(-90)))/12 b = (-12 +pm- sqrt(144-4(6)(-90)))/12 4*6*-90 = 4*6*-9*10 = 24*-9*10 = -216*10 = 2160 b = (-12 +pm- sqrt(144-(-2160)))/12 b = (-12 +pm- sqrt(2304))/12 b = -1 +pm- sqrt(2304)/12 sqrt(2304) = sqrt(4)*sqrt(576) = sqrt(4)*sqrt(4)*sqrt(144) = 2*2*12 = 48 b= -1 +pm- 48/12 = -1 +pm- 4 b=3 & b=-5 Solving for b=3: (2+3)^3+(3)^3 = 2x (5)^3+(3)^3 = 2x 125 + 27 = 2x 152 = 2x 76 = x Solving for b=-5: (2-5)^3+(-5)^3 =2x (-3)^3 + (-5)^3 = 2x -27 + -125 = 2x -152 = 2x -76 = x x = +pm- 76

@user-sk9oi9jl2g 8 ай бұрын

You can cube the right and left sides of the equation. After the transformations, we get the equation x^2-49=15^3.

@loufabellon8475 8 ай бұрын

Nice voice! Your videos are always soo nice!

@schlingel0017 5 ай бұрын

Solved in a similar way but I cubed the equation to get 98-6ab=8 and ab=15.

@avalagum7957 6 ай бұрын

Didn't know how to solve it until I saw you writing a = (x + 49)^(1/3) and b = (x - 49)^(1/3). Nice idea.

@user-do9mh4mh6z 2 ай бұрын

Я решала а=х+49, в=х-49

@monicagarg2948 8 ай бұрын

Nice bro and thanks

@learncommunolizer 8 ай бұрын

Welcome

@user-ec5ip3vp2r 2 ай бұрын

-76;76

@sauronbadeye 8 ай бұрын

Speaking a language of his own!!! ☹️☹️

@user-xh3ih4ks9y 8 ай бұрын

X=76

@user-im2qo6hh5e 5 ай бұрын

Я до сих пор не понимаю какими блудными путями у вас под корнями минус и как вы его достаете оттуда. Оставьте один ответ 76. Или давайте поговорим о мнимой единице и что такое модуль. Ну или, если я не прав, то поправьте кто-нибудь.

@user-do9mh4mh6z 2 ай бұрын

Х=76

@roger7341 7 ай бұрын

By inspection I zeroed in on 5 - 3 = 2, from which ∛125 - √27 = 2. This gives x + 49 = 125 and x - 49 = 27, so x = 76.

@bobwineland9936 8 ай бұрын

Extremely long and complicated process. I don't think i benefited from it but I'm impressed with your knowledge sir!

@percy9228 8 ай бұрын

all he had to do was cube both sides , and he would have got a quadratic equation which we can solve. just start of with the original equation and cube both sides ((x+49)^(1/3))^3 -3.(((x+49)(x-49))^(1/3))(2) -((x-49)(1/3))^3 = 2^3 x+49 -6((x+49)(x-49))^(1/3) -x+49 = 15 ((x+49)(x-49))^(1/3) = 15

@jimkekoa2756 6 ай бұрын

You could have solved this by inspection rather than going through all this complicated Algebra. This was something my 9th grade Algebra teacher taught us and I have used over the years. Why make your life hard?