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Math Help - Need help for Proving Quadratic Equations

  1. #1
    Junior Member
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    Post Need help for Proving Quadratic Equations

    Hello people,

    How do i prove these?

    1)
    If a, b and c are real, show that the roots of the following equations are real:

    (i) (b+c)x^2 - (a + b + c)x + a = 0
    (ii) (a + b)(x^2 + 5x + 1) = c(x^2-1)

    2)
    If roots of the equation (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a) = 0 are equal, prove that a = b = c.

    3) If a = b = c, prove that the roots of the equation:

    (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a) = 0 are equal.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Medgidia, Romania
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    1)
    i) The discriminant is

    \Delta=(a+b+c)^2-4a(b+c)=a^2+b^2+c^2-2ab-2ac+2bc=(a-b-c)^2\geq 0, so the roots are real

    ii) The equation is (a+b-c)x^2+5(a+b)x+a+b+c=0

    \Delta=25(a+b)^2-4(a+b+c)(a+b-c)=25(a+b)^2-4(a+b)^2+4c^2=21(a+b)^2+4c^2\geq 0

    2) The equation is

    3x^2-2(a+b+c)x+ab+ac+bc=0

    If the roots are equal then the discriminant is 0.

    \Delta=4(a+b+c)^2-12(ab+ac+bc)=4(a^2+b^2+c^2-ab-ac-bc)=

    =2[(a-b)^2+(a-c)^2+(b-c)^2]

    \Delta=0\Rightarrow a-b=0, \ a-c=0, \ b-c=0\Rightarrow a=b=c

    3) a=b=c\Rightarrow3(x-a)^2=0\Rightarrow x_1=x_2=a
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  3. #3
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    Cool
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