If a2+4 b2+9 c2+2 a b+6 b c+3 a c ≤ 0 where a, b, c ∈ R, then the number of distinct real roots of a x2+b x+c=0 will be

Question

If a2+4 b2+9 c2+2 a b+6 b c+3 a c ≤ 0 where a, b, c ∈ R, then the number of distinct real roots of a x2+b x+c=0 will be (A) 0 (B) 1 (C) 2 (D) more than 2

Tardigrade Admin · Accepted Answer

Θ 2 a2+8 b2+18 c2+4 a b+12 b c+6 a c ≤ 0 ⇒( a +2 b )2+(2 b +3 c )2+(3 c + a )2 ≤ 0 ⇒ a +2 b =0 ; 2 b +3 c =0 text and 3 c + a =0 ⇒ a = b = c =0 ∴ Equation ax 2+ bx + c =0 will be an identity

Q. If $a^{2} + 4 b^{2} + 9 c^{2} + 2 ab + 6 b c + 3 a c \leq 0$ where $a, b, c \in R$ , then the number of distinct real roots of $a x^{2} + b x + c = 0$ will be

0

1

2

more than 2

$Θ2 a^{2} + 8 b^{2} + 18 c^{2} + 4 ab + 12 b c + 6 a c \leq 0$
$\Rightarrow (a + 2 b)^{2} + (2 b + 3 c)^{2} + (3 c + a)^{2} \leq 0$
$\Rightarrow a + 2 b = 0; 2 b + 3 c = 0 and 3 c + a = 0$
$\Rightarrow a = b = c = 0$
$∴$ Equation $a x^{2} + b x + c = 0$ will be an identity

Q. If a2+4b2+9c2+2ab+6bc+3ac≤0 where a,b,c∈R, then the number of distinct real roots of ax2+bx+c=0 will be

0

1

2

more than 2

Θ2a2+8b2+18c2+4ab+12bc+6ac≤0 ⇒(a+2b)2+(2b+3c)2+(3c+a)2≤0 ⇒a+2b=0;2b+3c=0 and 3c+a=0 ⇒a=b=c=0 ∴ Equation ax2+bx+c=0 will be an identity

Q. If $a^{2} + 4 b^{2} + 9 c^{2} + 2 ab + 6 b c + 3 a c \leq 0$ where $a, b, c \in R$ , then the number of distinct real roots of $a x^{2} + b x + c = 0$ will be

$Θ2 a^{2} + 8 b^{2} + 18 c^{2} + 4 ab + 12 b c + 6 a c \leq 0$
$\Rightarrow (a + 2 b)^{2} + (2 b + 3 c)^{2} + (3 c + a)^{2} \leq 0$
$\Rightarrow a + 2 b = 0; 2 b + 3 c = 0 and 3 c + a = 0$
$\Rightarrow a = b = c = 0$
$∴$ Equation $a x^{2} + b x + c = 0$ will be an identity