500 followers problem!!!

Algebra Level 4

Given the equation

3 x 3 + 2 x 2 + x + 5 = 0 3x^3+2x^2+x+5=0

Has n n real roots and the sum of all the real roots is l l

Determine the value of

n [ l ] n-[l]


Where [...] is the GIF of ...


The answer is 3.

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2 solutions

Pranjal Jain
Jan 23, 2015

f ( x ) = 3 x 3 + 2 x 2 + x + 5 f(x)=3x^3+2x^2+x+5

g ( x ) = f ( x ) = 9 x 2 + 4 x + 1 g(x)=f'(x)=9x^2+4x+1

Now g ( x ) g(x) is always positive as coefficient of x 2 > 0 x^2>0 and < 0 ∆<0

Therefore, f ( x ) f(x) is strictly increasing and hence n = 1 n=1 .

  • f ( 0 ) = 5 > 0 f(0)=5>0
  • f ( 1 ) = 3 > 0 f(-1)=3>0
  • f ( 2 ) = 13 < 0 f(-2)=-13<0

Thus f ( x ) = 0 f(x)=0 has a solution between x = 2 x=-2 and x = 1 x=-1 .

l = 2 \Rightarrow \lfloor l\rfloor=-2

n l = 1 + 2 = 3 n-\lfloor l \rfloor=1+2=3

I did the same way . Good work. Upvoted :D

Keshav Tiwari - 6 years, 4 months ago

I like your solution. Congratulations.

Niranjan Khanderia - 6 years, 4 months ago

Nice solution

Paola Ramírez - 6 years, 4 months ago

Yuppp did the same way

Rajat Bisht - 6 years, 4 months ago

@Parth Lohomi I am really confused why does your problems move to level 5 and so less people can solve them. Was this worth 300+ points? 😕

Pranjal Jain - 6 years, 4 months ago

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Really can you solve all my level 5 problems!?? 😕

Parth Lohomi - 6 years, 4 months ago

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Hey! Cmon! I am not saying your problems are bad or something like that! They are nice! But when I solved it, it was worth 300+ points! And I was really surprised! Usually I have to work hard to get 300+ points' problem correct. Still let it be! Its down to 250 now. Keep up the hard work. ☺☺

Pranjal Jain - 6 years, 4 months ago

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@Pranjal Jain Is my solution correct ? I doubt !

Nihar Mahajan - 6 years, 4 months ago

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@Nihar Mahajan

  • If you are given a particular cubic equation, you must check if it has 3 roots or not. Maybe it has only 1 root. Though in general, a cubic can have maximum of 3 roots.

  • a + b + c ( a + b + c ) \lfloor a\rfloor+\lfloor b\rfloor+\lfloor c\rfloor ≠\lfloor (a+b+c)\rfloor . A counterexample can be 1.2 , 1.5 , 2.9 1.2,1.5,2.9

  • Pranjal Jain - 6 years, 4 months ago

    Obviously I can't solve all of your problems! 😅

    Pranjal Jain - 6 years, 4 months ago

    I met your brother vatsal today! ! I even know your house!!!! I Live very close to you,, in gopal vihar @Pranjal Jain

    Parth Lohomi - 6 years, 4 months ago

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    Oh I see! So many Brilliantians are from Kota!

    Pranjal Jain - 6 years, 4 months ago

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    @Pranjal Jain I think you are right @Pranjal Jain

    Rajat Bisht - 6 years, 4 months ago

    You can not say that!😡😡😬😬😠!

    Parth Lohomi - 6 years, 4 months ago
    Chew-Seong Cheong
    Jan 23, 2015

    A plot of the curve clearly shows that n = 1 n =1 and l = 2 \lfloor l \rfloor = -2 . Therefore, n l = 3 n - \lfloor l \rfloor = \boxed{3} .

    This is the best one !!!!!!

    Shubhendra Singh - 6 years, 4 months ago

    Please tell me how to plot this

    Mehul Chaturvedi - 6 years, 4 months ago

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    I will put this in a note.

    Chew-Seong Cheong - 6 years, 4 months ago

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    sir, where is that note?

    Dev Sharma - 5 years, 7 months ago

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