Quadratic Function

  1. [SOLVED] Quadratic Function

    1. The problem statement, all variables and given/known data


    solve (f(x)=-3 x^2 + 9 x + 1/4 )
    fx=(-3)(x^2-3 x)+1/4

    fx=(-3)(x^2-3 x+9/4)+1/4+27/4

    fx=(-3)(x^2-3 x+9/4)+28/4
    fx=(-3) (x-3/2)^2+7



    2. Relevant equations



    3. The attempt at a solution


    the solution is already given in the book, but I don't understand why 9/4 was inserted in the third line (bolded). I know at the end of the equation, 27/4 is added to balance out the 9/4.......but why put in 9/4 there?
     
  2. jcsd
  3. Kurdt

    Kurdt 4,941
    Staff Emeritus
    Science Advisor
    Gold Member

    What they've done is simply completed the square. If you have an equation of the form: [itex] x^2+2ax [/itex], you can write it as: [itex] x^2+2ax = (x+a)^2 - a^2 [/itex].
     
    Last edited: Nov 28, 2007
  4. rock.freak667

    rock.freak667 6,220
    Homework Helper

    f(x)=-3 x^2 + 9 x + 1/4

    they just put 27/4 and -27/4 to make completion easier

    f(x)=-3 x^2 + 9 x -27/4+ 1/4 +27/4
    =-3(x^2-3x+9/4)+7
    =-3(x-3x/2)^2 +7

    expand (x-3x/2)^2 and check it yourself
     
  5. yea bro...

    the thing is, how do you come up with 9/4 from to make it easier?

    I'm on purplemath.com right now, trying to figure it out without bothering you guys... hopefully I can figure this one out.
     
  6. Kurdt

    Kurdt 4,941
    Staff Emeritus
    Science Advisor
    Gold Member

    Well heres how I would have done that problem. You're given:

    [tex] f(x) = -3x^2 + 9x + \frac{1}{4} [/tex]
    [tex] f(x) = -3( x^2 -3x) + 1/4 [/tex]

    Now we notice that the term in brackets is of the form [itex] x^2+2ax [/itex], with [itex]a=\frac{-3}{2}[/itex], and so we complete the square [itex] x^2+2ax = (x+a)^2 - a^2 [/itex]:

    [tex] f(x) = -3\left(\left(x-\frac{3}{2}\right)^2 - \frac{9}{4} \right) + \frac{1}{4} [/tex]
    [tex] f(x) = -3\left(x - \frac{3}{2}\right)^2 + \frac{27}{4} + \frac{1}{4} [/tex]
    [tex] f(x) = -3\left(x - \frac{3}{2}\right)^2 + 7 [/tex]
     
    Last edited: Nov 28, 2007
  7. okie.........i got it.


    so the point of adding -9/4 inside the brackets is so that you can make it factorable. And since you added a (-), now you have to minus it to even it out.


    thnkx.....can't believe it took me so long.

    anyhow, how do you type the math out like that?
     
    Last edited: Nov 28, 2007
  8. test:

    [tex]x=-3 x^2+9x+1/4[/tex]

    [tex]fx=(-3)(x^2┤-3 x)+1/4\\[/tex]

    [tex]fx=(-3)(x^2┤-3 x+9/4)+1/4+27/4\\[/tex]

    [tex]fx=(-3)(x^2┤-3 x+9/4)+28/4\\[/tex]

    [tex]fx=(-3) (x┤-3/2)^2+7\\[/tex]
     
    Last edited: Nov 29, 2007
  9. Kurdt

    Kurdt 4,941
    Staff Emeritus
    Science Advisor
    Gold Member

    If you want to test then there are plenty of preview websites. Here is one of them.

    http://at.org/~cola/tex2img/index.php
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook