Results 1 to 4 of 4
Like Tree3Thanks
  • 1 Post By Trefoil2727
  • 1 Post By Prove It
  • 1 Post By Soroban

Math Help - Algberic question

  1. #1
    Newbie
    Joined
    Jun 2013
    From
    india
    Posts
    16

    Algberic question

    If a^4+a^2 b^2 + b^4 = 8 and a^2 + ab + b^2 = 4, then the value of ab is

    (A) -1 (B) 0 (C) 2 (D) 1
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Apr 2013
    From
    Mahinog
    Posts
    81
    Thanks
    5

    Re: Algberic question

    Quote Originally Posted by sscgeek View Post
    If a^4+a^2 b^2 + b^4 = 8 and a^2 + ab + b^2 = 4, then the value of ab is

    (A) -1 (B) 0 (C) 2 (D) 1
    a^2 + b^2 =4-ab
    (a+b)^2 = 4+ ab...(1)

    a^4 + b^4 = 8- (ab)^2
    (a+b)^2 = 2ab+ √ (8+ (ab)^2)...(2)

    4+ab= 2ab +√ (8+ (ab)^2)
    4-ab= √ (8+ (ab)^2)
    16-8ab + (ab)^2 = 8+ (ab)^2
    8ab + 8
    ab=1
    Last edited by Trefoil2727; June 16th 2013 at 05:19 AM. Reason: typing error
    Thanks from HallsofIvy
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,404
    Thanks
    1293

    Re: Algberic question

    From the first equation:

    \displaystyle \begin{align*} a^4 + a^2b^2 + b^4 &= 8 \\ \left( a^2 \right) ^2 + 2a^2b^2 + \left( b^2 \right) ^2 - a^2b^2 &= 8 \\ \left( a^2 + b^2 \right) ^2 - a^2b^2 &= 8 \end{align*}

    and from the second equation:

    \displaystyle \begin{align*} a^2 + a\,b + b^2 &= 4 \\ a^2 + b^2 &= 4 - a\,b \\ \left( a^2 + b^2 \right) ^2 &= \left( 4 - a\,b \right) ^2 \\ \left( a^2 + b^2 \right) ^2 &= 16 - 8\,a\,b + a^2b^2 \end{align*}

    Substituting into the first equation gives

    \displaystyle \begin{align*} 16 - 8\,a\,b + a^2b^2 - a^2b^2 &= 8 \\ 16 - 8\,a\,b &= 8 \\ -8\,a\,b &= -8 \\ a\,b &= 1 \end{align*}
    Thanks from HallsofIvy
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,678
    Thanks
    611

    Re: Algberic question

    Hello, sscgeek!

    \text{Given: }\:\begin{Bmatrix}a^4+a^2 b^2 + b^4 &=& 8 & [1] \\ a^2 + ab + b^2 &=& 4 & [2]\end{Bmatrix}

    \text{Find the value of }ab.

    . . (A)\;-1 \qquad (B)\;0 \qquad (C)\;2 \qquad (D)\;1


    \begin{array}{ccccccc}\text{Square [2]:} & a^4 + 2a^3b + 3a^2b^2 + 2ab^3 + b^4 &=& 16 \\ \text{Subtract [1]:} & a^4\quad\;\;+ \quad\;\; a^2b^2 \quad\;\; + \quad\;\; b^4 &=&  8 \end{array}

    \text{We have: }\;\; 2a^3b + 2a^2b^2 + 2ab^2 \;=\;8

    . . . . . . . . . . ab\underbrace{(a^2 + ab + b^2)}_{\text{This is 4}} \;=\; 4

    . . . . . . . . . . . . . . . . . ab(4) \;=\;4

    . . . . . . . . . . . . . . . . . . . ab \;=\;1 \;\;\text{ answer (D)}
    Thanks from HallsofIvy
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum