How to do this C3 question?

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  1. Fusionary's Avatar
    1 13-06-2012  15:08
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    How to do this C3 question?
    1 ) The functions f and g are defined for real values of x by

    f(x) = 4x^2 -12x , g(x)= ax+ b

    Where a and b are non-zero constants.

    Given that a = -1 and given further that gf(x)<5 , for all values of x, find the set of possible values of b.
    Last edited by Fusionary; 13-06-2012 at 15:10.
  2. Fusionary's Avatar
    2 13-06-2012  15:11
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    Re: How to do this C3 question?
    Please post a detailed worked solution in spoiler, thanks.
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  3. BabyMaths's Avatar
    3 13-06-2012  15:11
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    Re: How to do this C3 question?
    Sub in what you're given and write the inequality. Post what you get.
  4. Fusionary's Avatar
    4 13-06-2012  15:14
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    Re: How to do this C3 question?
    -(4x^2-12x) + b < 5
  5. james22's Avatar
    5 13-06-2012  15:15
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    Please post a detailed worked solution in spoiler, thanks.
    We aren't supposed to give full solutions until we can see that you have had a good go at it. Post what you have done so far.
  6. Fusionary's Avatar
    6 13-06-2012  15:16
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    Re: How to do this C3 question?
    -(4x^2 - 12x) + b < 5 \rightarrow -4x^2 + 12x + b <5

    Now?
  7. james22's Avatar
    7 13-06-2012  15:17
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    -(4x^2-12x) + b < 5
    You have be told that this is true for all x. I would now put it all on one side to give you a quadratic >0 or <0 then complete the square.
  8. Fusionary's Avatar
    8 13-06-2012  15:19
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    Re: How to do this C3 question?
    -4x^2 + 12x + b - 5 &lt;0

    Not sure how to complete the square for this?
    Last edited by Fusionary; 13-06-2012 at 15:23.
  9. james22's Avatar
    9 13-06-2012  15:20
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    4x^2 + 12x + b - 5 &lt;0

    Not sure how to complete the square for this?
    Do you know how to complete the square for one with just numbers in? If so it's exactly the same, just treat the final contant as (b-5) and complete the square as normal.
  10. Mally1's Avatar
    10 13-06-2012  15:22
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    Re: How to do this C3 question?
    Good luck for C3!!!
    My paper last year in June was lovely, I hope the same for you guys : )
  11. raheem94's Avatar
    11 13-06-2012  15:22
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    4x^2 + 12x + b - 5 &lt;0

    Not sure how to complete the square for this?
    Shouldn't the first term be - 4x^2 \ ?
  12. Fusionary's Avatar
    12 13-06-2012  15:23
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    Re: How to do this C3 question?
    Hmm, yes it should, thank you
    Last edited by Fusionary; 13-06-2012 at 15:24.
  13. Fusionary's Avatar
    13 13-06-2012  15:26
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    Re: How to do this C3 question?
    Okay so -(4x^2-12x) + b - 5 &lt; 0 Therefore \rightarrow -(2x-6)^2 + 36 + b - 5 &lt; 0 so \rightarrow -(2x-6)^2 + 31 + b &lt; 0 and now?
    Last edited by Fusionary; 13-06-2012 at 15:28.
  14. james22's Avatar
    14 13-06-2012  15:30
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    Okay so -(4x^2-12x) + b - 5 &lt; 0 Therefore \rightarrow -(2x-6)^2 + 36 + b - 5 &lt; 0 so \rightarrow -(2x-6)^2 + 31 + b &lt; 0 and now?
    You've gone wrong somewere, it should be 2x-3 in the bracket not 2x-6. It would be much easier to do if you divided everything by -4 then completed the square. You can always multiply by 4 again afterwards to make the numbers easier.
  15. raheem94's Avatar
    15 13-06-2012  15:30
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    Okay so -(4x^2-12x) + b - 5 &lt; 0 Therefore \rightarrow -(2x-6)^2 + 36 + b - 5 &lt; 0 so \rightarrow -(2x-6)^2 + 31 + b &lt; 0 and now?
    Is the answer b &lt; -4 \ ?
  16. Fusionary's Avatar
    16 13-06-2012  15:32
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    Re: How to do this C3 question?
    -(2x-3)^2 + 4 + b &lt; 0
  17. Fusionary's Avatar
    17 13-06-2012  15:32
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    Re: How to do this C3 question?
    (Original post by raheem94)
    Is the answer b &lt; -4 \ ?
    yes it is, can you post a worked solution in spoiler please?
  18. raheem94's Avatar
    18 13-06-2012  15:33
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    Okay so -(4x^2-12x) + b - 5 &lt; 0 Therefore \rightarrow -(2x-6)^2 + 36 + b - 5 &lt; 0 so \rightarrow -(2x-6)^2 + 31 + b &lt; 0 and now?
    -(4x^2-12x) + b - 5 &lt; 0 \implies 4x^2 - 12x + (5-b) &gt; 0

    Hence the equation has no real roots, consider the discriminant, b^2 &lt; 4ac

    b^2 &lt; 4ac \implies 144 &lt; 4 \times 4 \times (5-b)
  19. james22's Avatar
    19 13-06-2012  15:33
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    Re: How to do this C3 question?
    That is correct, now since this is true for all x, can you find the values fo b? What is (2x-3)^2 always bigger than?
  20. raheem94's Avatar
    20 13-06-2012  15:34
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    Re: How to do this C3 question?
    (Original post by Fusionary)
    yes it is, can you post a worked solution in spoiler please?
    I have posted some working, if it doesn't makes sense tell me, so that i can explain it further.
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