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The discriminant of a quadratic and real solutions

  1. May 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Given a series of mathematical statements, some of which are true and some of which are false. Prove or Disprove:
    1. A sufficient condition that ax2+bx+c=0 (a≠0) have a real root is that b2-4ac>5.
    2.A necessary condition that ax2+bx+c=0 (a≠0) have a real root is that b2-4ac=0.

    2. Relevant equations
    x(px→qx)


    3. The attempt at a solution
    1.
    A sufficient condition that ax2+bx+c=0 (a≠0) have a real root is that b2-4ac>5.

    If b2-4ac>5, then ax2+bx+c=0 (a≠0) has a real root.
    ax2+bx+c=0
    x2+(b/a)x+(c/a)=0
    x2+(b/a)x+(b2/4a2)+(c/a)=(b2/4a2)
    (x+(b/2a))2+(c/a)=(b2/4a2)
    (x+(b/2a))2=(b2/4a2)-(c/a)
    (x+(b/2a))2=(b2/4a2)-(4ac/4a2)
    x+(b/2a)=(±√b2-4ac)/2a
    x=(-b±√b2-4ac)/2a
    ±√b2-4ac≥0
    b2-4ac≥0
    Since 5>0, b2-4ac>5 is sufficient condition for the equation to a have real root.

    2. A necessary condition that ax2+bx+c=0 (a≠0) have a real root is that b2-4ac=0.

    If and only if b2-4ac=0, then ax2+bx+c=0 (a≠0) has a real root.
    From 1. we know that ax2+bx+c=0 (a≠0) has a real root when b2-4ac>5.
    Therefore it is false that the equation has a real root if and only if the discriminant is zero.
     
  2. jcsd
  3. May 30, 2015 #2

    LCKurtz

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    For part 2 you are only asked whether the discriminant being zero is necessary to have a real root, not whether it is also sufficient (which it is). So you should only be concerned with the "only if" statement. I would leave out the red words.
     
  4. May 30, 2015 #3
    Gotcha', thanks.
     
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