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Math Help - What's the "best" Way of Solving Trinomial with leading coefficient that's not one?

  1. #1
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    What's the "best" Way of Solving Trinomial with leading coefficient that's not one?

    By best I mean best for you. I posted this thread to find out different ways people use. I know what I've been taught is popular according to my teacher.

    My way is:
    Example:

    3a^2+10a+7

    write the first and last numbers. The middle number "10a" is what we will divide into two numbers.

    3a^2+ ______+_______+7

    Multiple 3 by 7= 21.

    Our goal is similar to a normal trinomial without a leading coefficient more than one now. We need to find a number that when multiplied is 21 and added 10. Eventually, the numbers 3 and 7 will come up. Better to go in chronological order for those who do not know.

    3a^2+3a+7a+7

    Now use grouping method.

    3a(a+1) 7(a+1)

    Take the numbers 3a and 7... put them in (3a+7)

    (a+1) (a+1) is similar to doing it with (3a+7)

    answer: (3a+7)(a+1)

    =(3a+7) (a+1)

    This is how I do it. It might not make sense for you, but it does for me.

    Now, what's your way?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: What's the "best" Way of Solving Trinomial with leading coefficient that's not on

    Quote Originally Posted by xivivx View Post
    By best I mean best for you. I posted this thread to find out different ways people use. I know what I've been taught is popular according to my teacher.

    My way is:
    Example:

    3a^2+10a+7

    write the first and last numbers. The middle number "10a" is what we will divide into two numbers.

    3a^2+ ______+_______+7

    Multiple 3 by 7= 21.

    Our goal is similar to a normal trinomial without a leading coefficient more than one now. We need to find a number that when multiplied is 21 and added 10. Eventually, the numbers 3 and 7 will come up. Better to go in chronological order for those who do not know.

    3a^2+3a+7a+7

    Now use grouping method.

    3a(a+1) 7(a+1)

    Take the numbers 3a and 7... put them in (3a+7)

    (a+1) (a+1) is similar to doing it with (3a+7)

    answer: (3a+7)(a+1)

    =(3a+7) (a+1)

    This is how I do it. It might not make sense for you, but it does for me.

    Now, what's your way?

    The method you described above called: Vieta's Formulas -- from Wolfram MathWorld
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  3. #3
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    Re: What's the "best" Way of Solving Trinomial with leading coefficient that's not on

    I've tried my method on 40 problems. I , however, got stuck at this one:

    3x^2-15x+16

    Why can't I solve this with the Vieta formula?


    Can you try and do it ?
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: What's the "best" Way of Solving Trinomial with leading coefficient that's not on

    Quote Originally Posted by xivivx View Post
    I've tried my method on 40 problems. I , however, got stuck at this one:

    3x^2-15x+16

    Why can't I solve this with the Vieta formula?


    Can you try and do it ?

    If \alpha and \beta are the two roots of the above equation, then:

    \alpha+\beta=5

    and,

    \alpha \cdot \beta=\frac{16}{3}

    solve that system of equations, and you will get your roots.
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  5. #5
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    Re: What's the "best" Way of Solving Trinomial with leading coefficient that's not on

    I can't find it. I think this darn thing is a prime. Try to find it -.-
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  6. #6
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    Re: What's the "best" Way of Solving Trinomial with leading coefficient that's not on

    Quote Originally Posted by Also sprach Zarathustra View Post
    \alpha+\beta=5 and,
    \alpha \cdot \beta=\frac{16}{3}

    solve that system of equations, and you will get your roots.
    That simply leads back to the original equation!

    Try the quadratic formula: b^2 - 4ac = 33 ; get my drift?
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