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Is there a more convenient way to solve quadratic trinomials with large coefficients?

  1. Feb 3, 2012 #1
    I'm told that factoring is an important skill in calculus so I am avoiding using the quadratic formula. But for quadratic equations with large coefficients to factor, is there a better/faster way rather than guessing and checking every single combination?
     
  2. jcsd
  3. Feb 3, 2012 #2

    Char. Limit

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    Re: Is there a more convenient way to solve quadratic trinomials with large coefficie

    I personally prefer completing the square. It's rather more useful in calculus than even factoring, as it gives you the point where the derivative is zero without even ever having to take it.

    EDIT:

    Completing the square is also rather simple. Start with an equation a x^2 + b x + c = 0, right? Divide both sides by a so that you get this:

    [tex]x^2 + \frac{b}{a} x + \frac{c}{a} = 0[/tex]

    Then add and subtract b^2 / 4 a^2 to the left side (effectively adding zero)

    [tex]x^2 + \frac{b}{a} x + \frac{b^2}{4 a^2} + \frac{c}{a} - \frac{b^2}{4 a^2} = 0[/tex]

    Now, the first three terms in that are a perfect square, and the last two are a constant, so we can easily rearrange this to give...

    [tex]\left( x + \frac{b}{2a}\right)^2 = \frac{b^2}{4 a^2} - \frac{c}{a}[/tex]

    And there we have it. This gives us the two roots with just a little algebraic manipulation (equivalent to the quadratic formula, as it happens) and also immediately gives us the vertex of the equation (hint, take the derivative of both sides).
     
  4. Feb 3, 2012 #3
    Re: Is there a more convenient way to solve quadratic trinomials with large coefficie

    Yes...the quadratic formula.
    How do you intend to find irrational roots without using the quadratic formula?
     
  5. Feb 3, 2012 #4

    Char. Limit

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    Re: Is there a more convenient way to solve quadratic trinomials with large coefficie

    If you take a look at my post above, it can be done quite easily.
     
  6. Feb 3, 2012 #5
    Re: Is there a more convenient way to solve quadratic trinomials with large coefficie

    True, but at that point he's effectively utilizing the quadratic formula anyway.
     
  7. Feb 3, 2012 #6

    Char. Limit

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    Re: Is there a more convenient way to solve quadratic trinomials with large coefficie

    Also true. I like completing the square better than a straight application, though, as it gives you more insight into why the quadratic formula works.
     
  8. Feb 3, 2012 #7

    mathman

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    Re: Is there a more convenient way to solve quadratic trinomials with large coefficie

    After doing it a few times your way the concept should sink in. After that using the quadratic formula is faster.
     
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