If the quadratic equations 3 x2+a x+1=0 and 2 x2+b x+1=0 have a common root, then the value of the expression 5 a b-2 a2-3 b2 is

Question

If the quadratic equations 3 x2+a x+1=0 and 2 x2+b x+1=0 have a common root, then the value of the expression 5 a b-2 a2-3 b2 is (A) 0 (B) 1 (C) -1 (D) 2

Tardigrade Admin · Accepted Answer

6 x2+2 a x+2=0 and 6 x2+3 b x+3=0 subtracting x(2 a-3 b)-1=0 ⇒ x=(1/2 a-3 b) (put in any equation) ∴ 2 (1/(2 a-3 b)2)+(b/2 a-3 b)+1=0 2+b(2 a-3 b)+(2 a-3 b)2=0 4 a2+5 b2-12 a b+2 a b-3 b2+2=0 -10 a b+6 b2+4 a2+1=0 ⇒ 5 a b-3 b2-2 a2=1 ⇒ B

Q. If the quadratic equations 3x2+ax+1=0 and 2x2+bx+1=0 have a common root, then the value of the expression 5ab−2a2−3b2 is

0

1

-1

2

6x2+2ax+2=0 and 6x2+3bx+3=0 subtracting x(2a−3b)−1=0⇒x=2a−3b1​ (put in any equation) ∴2(2a−3b)21​+2a−3bb​+1=0 2+b(2a−3b)+(2a−3b)2=0 4a2+5b2−12ab+2ab−3b2+2=0 −10ab+6b2+4a2+1=0 ⇒5ab−3b2−2a2=1⇒B

Q. If the quadratic equations $3 x^{2} + a x + 1 = 0$ and $2 x^{2} + b x + 1 = 0$ have a common root, then the value of the expression $5 ab - 2 a^{2} - 3 b^{2}$ is