Question

If $a,b,c$ are real numbers satisfying the condition $a+b+c=0,$ then the roots of the quadratic equation $3ax^{2}+5bx+7c=0$ are

• A. Positive
• B. negative
• C. real and equal
• D. distinct but not imaginary

Answer 1

Answer: (D)

$D=25b^{2}-4\times 3a\times 7c$
$=25\left(- a - c\right)^{2}-84ac$
$=25\left(a^{2} + c^{2} + 2 a c\right)-84ac$
$=25\left(a^{2} + c^{2}\right)-34ac$
$=17\left(a^{2} + c^{2} - 2 a c\right)+8\left(a^{2} + c^{2}\right)$
$=17\left(a - c\right)^{2}+8\left(a^{2} + c^{2}\right)$
$D>0\Rightarrow $ Roots are real and distinct