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How to determine the roots of a quadratic equation

  1. Jan 29, 2006 #1
    Hello All

    I am trying to figure out how to determine if there are two, one, zero, or infinitely many roots for given formula ax^2 + bx + c. Are there any easy ways to determine this with out having to use the quadratic formula?

    Thanks

    P
     
  2. jcsd
  3. Jan 29, 2006 #2

    Tide

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    I believe you are asking whether the polynomial has real roots. You can determine that by defining a function

    [tex]f(x) = ax^2 + bx + c[/tex]

    and completing the square

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

    The graph of x is a parabola whose vertex is at

    [tex]x = -\frac {b}{2a}[/tex]

    Given the sign of a you can determine whether the vertex is a maximum or a minimum and determine whether f(x) = 0 is possible.
     
  4. Jan 29, 2006 #3

    Well if you're referring to roots in a general sense then yes there is an easy way, it's called the fundamental theorem of algebra. A polynomial of degree n has exactly n roots including multiplicities.
     
  5. Jan 29, 2006 #4

    0rthodontist

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    Another way, maybe the way in your book, is using the discriminant b^2 - 4ac. If that's greater than 0 then the equation has 2 real roots, if it's less than 0 the equation has 0 real roots, and if it is equal to 0 then the equation has 1 real root. It comes out of the quadratic formula but you don't need to use the entire formula, just the part under the square root sign.
     
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