If a,b,c are real numbers satisfying the condition a+b+c=0, then the roots of the quadratic equation 3ax2+5bx+7c=0 are

Question

If a,b,c are real numbers satisfying the condition a+b+c=0, then the roots of the quadratic equation 3ax2+5bx+7c=0 are (A) Positive (B) negative (C) real and equal (D) distinct but not imaginary

Tardigrade Admin · Accepted Answer

D=25b2-4× 3a× 7c =25(- a - c)2-84ac =25(a2 + c2 + 2 a c)-84ac =25(a2 + c2)-34ac =17(a2 + c2 - 2 a c)+8(a2 + c2) =17(a - c)2+8(a2 + c2) D>0⇒ Roots are real and distinct

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

Positive

negative

real and equal

distinct but not imaginary

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

Q. If a,b,c are real numbers satisfying the condition a+b+c=0, then the roots of the quadratic equation 3ax2+5bx+7c=0 are