Question

(a) Discuss the nature of the roots of the equations $4a{x}^2+3bx+2c=0$ where $a,b,cϵR$ and are connected by the relation $a+b+c=0$.
(b) If the roots of the equation
$[a^2+291-b)]x^2+2a(1+b)x+2b(b-1)+a^2=0$ be equal, then prove that $a^2=4b$.

Answer 1

(a) $\Delta =9b^2-32ac=9b^2+32a(a+b)$
$=9b^2+32ab+32a^2=b^2+8(b^2+4ab+4a^2)$
$=b^2+8{(b+2a)}^2=+ive$ $\therefore$ Real
(b) $\Delta =0$
$\Rightarrow 4a^2{(1+b)}^2-4[a^2+2b(b-1\left]\right[a^2-2(b-1)]=0$
or $a^2\{{(b-1)}^2+4b\}-\{a^4+2a^2(b^2-2b+1)-4b{(b-1)}^2\}=0$
or $a^4+2a^2{(b-1)}^2-4b{(b-1)}^2-a^2{(b-1)}^2-4b{a}^2=0$
or $a^2(a^2-4b)+{(b-1)}^2\{a^2-4b\}=0$
or $(a^2-4b)[a^2+{(b-1)}^2]=0$
$\therefore a^2-4b=0$ as the other factor cannot be zero.