1. Tardigrade
  2. Question
  3. Mathematics
  4. Statement-1: If a + b + c >0 and a <0< b < c, then both roots of the quadratic equation a(x-b)(x-c)+b(x-c)(x-a)+c(x-a)(x-b)=0 are real and unequal. Statement-2: If both roots of the quadratic equation px 2+ qx + r =0 are of opposite sign then product of roots is negative and sum of roots is positive.

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Q. Statement-1: If $a + b + c >0$ and $a <0< b < c$, then both roots of the quadratic equation $a(x-b)(x-c)+b(x-c)(x-a)+c(x-a)(x-b)=0$ are real and unequal.
Statement-2: If both roots of the quadratic equation $px ^2+ qx + r =0$ are of opposite sign then product of roots is negative and sum of roots is positive.

Complex Numbers and Quadratic Equations

Solution:

We have $(a+b+c) x^2-2(a b+b c+c a) x+3 a b c=0$
$\therefore $ Discriminant $=D=4\left[(a b+b c+c a)^2-3(a+b+c) a b c\right] ($ Since $a+b+c>0$ and $a b c<0)$
$\Rightarrow D >0$
Hence roots are real and distinct.
Product of roots $=\frac{3 a b c}{a+b+c}<0$
$\Rightarrow $ Roots are of opposite sign.
Statement - 1 is True and Statement - 2 is False.