Question

If 1 is a root of the quadratic equation $3x^2$ + ax - 2 = 0 and the quadratic equation a($x^2$ + 6x) - b = 0 has equal roots, The value of b is -------.

Answer 1

$3x^2+ax-2=0$

Since, one root is $1$, then $x=1$

$\Rightarrow$ $3(1{)}^2+a(1)-2=0$

$\Rightarrow$ $3+a-2=0$

$\Rightarrow$ $a+1=0$

$\Rightarrow$ $a=-1$

$\Rightarrow$ Now, it is given that $a{x}^2+6ax-b=0$ has equal roots.

$\therefore$ $b^2-4ac=0$

$\Rightarrow$ $(6a{)}^2-4(a\left)\right(-b)=0$

$\Rightarrow$ $36a^2+4ab=0$

$\Rightarrow$ $36(-1{)}^2+4(-1)b=0$ [ Substituting $a=-1$ ]

$\Rightarrow$ $36-4b=0$

$\Rightarrow$ $4b=36$

$\therefore$ $b=9$