Smart Method: Solving Without Squaring First | Math Olympiad

[@DrWangUSA]

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[@festel1432]

You can notice that the left and right parts of the equation are one function with different arguments. f(t) = sqrt(t) + sqrt(t + 5). This function is monotonically increasing, therefore it takes each of its values at one point. f(a) = f(b), where a = x + 2, b = 3x + 1 in the only way when a is b. x + 2 = 3x + 1; 2x = 1; x = 0.5

[@DrWangUSA]

Nice work and thanks for sharing

[@theupson]

you could have exploited the central observation more neatly. U^.5+ (U-5)^.5 is monotonic in U and therefore U^.5+(U-5)^.5 = V^.5 +(V-5)^.5 implies U=V.

[@DrWangUSA]

Thanks for sharing

[@manuelgonzales2570]

Very interesting. Thank you!

[@math_qz_2]

Nice solution 🙂 I like it

[@DrWangUSA]

Thanks

[@dlp778]

Nice solution ❤

[@DrWangUSA]

Thanks

[@師太滅絕]

I cannot follow this solution. A - B = C - D (original where all A, B, C, and D are square-root) A - B = C - D cannot, and does not makes A^2 - B^2 = C^2 - D^2

[@DrWangUSA]

Watch the video again you may see why.

[@mathsfamily6766]

nice method!

[@DrWangUSA]

Thanks

[@Simonblaze420]

You can actually see that x must be 1/2 just by comparing both sides

[@DrWangUSA]

Thanks for sharing

[@wesleydeng71]

Notice if x+7=3x+6 then x+2=3x+1, also if x+7>3x+6 then x+2>3x+1 and vice versa. So the only solution is when x+7=3x+6. x=1/2.

[@DrWangUSA]

Thanks for sharing

[@9허공]

given sqrt( x + 7) + sqrt( x + 2 ) = sqrt( 3x + 6 ) + sqrt( 3x + 1 ) sqrt( x + 7 ) - sqrt( 3x + 1 ) = sqrt( 3x + 6 ) - sqrt( x + 2 ) sq both sides, 4x + 8 - 2 * sqrt( ( x + 7 ) * ( 3x + 1 ) = 4x + 8 - 2 * sqrt( ( 3x + 6) * (x + 2) ) 3 * x^2 + 22x + 7 = 3 * x^2 +12x + 12 => x = 1/2

[@DrWangUSA]

Thanks for sharing

[@nasrullahhusnan2289]

Let u=x+2 and v=3x+1 Then (u+5)^½+(u^½)=(v+5)^½+(v^½) --> u=v x+2=3x+1 --> x=½

[@DrWangUSA]

Nice work and thanks for sharing

[@joeljohn839]

Why not use logarithms which gives 1/2 and -2 as solutions and when checked 1/2 can be selected

[@DrWangUSA]

It would be great if you can share details with us.

[@StephenRayWesley]

7x+2x=,7x-2x=14x+4x=√18x√ 3^√6 √1^3^2 3^2 (x+2x-3) 3x+6x=3x-6x=6x+,12x=18x +3x+1x=3x-1x=6x+2x=,8x (18x+8x)=√26x 2^√13. 2^1 (x+1x-2)

[@DrWangUSA]

Thanks for sharing

[@ZipplyZane]

Hope you guys are okay! Haven't heard from you in over a month.

[@SrisailamNavuluri]

If x+7=3x+6,2x=,x=1/2 If x+2=3x+1,2x=1,x=1/2 So x=1/2.

[@DrWangUSA]

Thanks for sharing

[@mustafasibic2954]

Thos is what happens when you don't know the trick x+2= t Sqrt(t+5) + sqrt(t) = sqrt(3t) + sqrt(3t-5) Square both sides 2t+5+ 2sqrt(t^2+5t)=6t-5+ 2sqrt(9t^2-15t) 2(sqrt(t^2+5t)-sqrt(9t^2-15t)= 4t-10 sqrt(t^2+5t)-sqrt(9t^2-15t)= 2t-5 sqrt(t^2+5t) = a, sqrt(9t^2-15t) = b Square again 10t^2-10t+2ab=4t^2-20t+25 6t^2+10t-25=-2ab Square again 36t^4+120t^3-200t^2-500t+625= 4(9t^4+30t^3-75t^2) 36t^4+120t^3-200t^2-500t+625 = 36t^4+ 120t^3- 300t^2 -200t^2-500t+625=-300t^2 100t^2+500t-625=0 4t^2+20t-25=0 (2t-5)^2=0 2t-5=0 2x-4+5=0 2×-1=0 x=1/2

[@DrWangUSA]

Nice work and thanks for sharing

[@walterwen2975]

Math Olympiad: √(x + 7) + √(x + 2) = √(3x + 6) + √(3x + 1); x = ? [√(x + 7) - √(3x + 1)]² = [√(3x + 6) - √(x + 2)]² (x + 7) - 2√[(x + 7)(3x + 1)] + (3x + 1) = (3x + 6) - 2√[(3x + 6)(x + 2)] + (x + 2) √[(x + 7)(3x + 1)] = √[(3x + 6)(x + 2)], (x + 7)(3x + 1) = (3x + 6)(x + 2) 3x² + 22x + 7 = 3x² + 12x + 12, 10x = 5; x = 1/2 Answer check: √(x + 7) + √(x + 2) = √(3x + 6) + √(3x + 1) √(1/2 + 7) + √(1/2 + 2) = √[(1 + 14)/2] + √[(1 + 4)/2] = √(15/2) + √(5/2) √(3/2 + 6) + √(3/2 + 1) = √[(3 + 12)/2] + √[(3 + 2)/2] = √(15/2) + √(5/2); Confirmed Final answer: x = 1/2

[@DrWangUSA]

Thanks for sharing