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Math Help - Real roots

  1. #1
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    Real roots

    Show that the equation x^2+(3a-2)x+a(a-1)=0 has real roots for all values of a are real numbers ..

    I tried this ..
    (3a-2)^2-4(a^2-a)
    =9a^2-12a+4-4a^2+4a
    =5a^2-8a+4

    Is it because 5a^2-8a+4 is a quadratic equation and hence it has real roots ? Or i am in the wrong path ...
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by mathaddict View Post
    Show that the equation x^2+(3a-2)x+a(a-1)=0 has real roots for all values of a are real numbers ..

    I tried this ..
    (3a-2)^2-4(a^2-a)
    =9a^2-12a+4-4a^2+4a
    =5a^2-8a+4

    Is it because 5a^2-8a+4 is a quadratic equation and hence it has real roots ? Or i am in the wrong path ...
    In order to prove that the first equation (x+(3a-2)x+a(a-1)=0) has real roots, you have to prove that the discriminant, 5a-8a+4 is always positive (or equal to 0).

    So you have to see when 5a^2-8a+4\geq 0
    In order to prove this, find the discriminant : 64-16*5<0
    Hence the sign of 5a-8a+4 is always the same, and its sign is the sign of the leading coefficient, 5.

    Thus 5a^2-8a+4>0 for all a.

    Therefore, the equation has real roots for all a.



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  3. #3
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    you would have to show that the descriminant is always greater or equal zero for a as real number, so you would have to show that 5a^2-8a+4 is always greater than or equal to zero. You could do this by plotting the function on a graph.
    sorry for repeating what was just said i think i posted it at the same time.
    Last edited by hmmmm; January 11th 2009 at 03:24 AM.
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