Question

If the quadratic equation $12x^2–2bx+9=0$ has no real root, then which of the following is always true?

• A. $b<-6\sqrt{3}$
• B. $b>6\sqrt{3}$
• C. $-6\sqrt{3}<b<6\sqrt{3}$
• D. $b<-6\sqrt{3}andb>6\sqrt{3}$

Answer 1

The correct option is C $-6\sqrt{3}<b<6\sqrt{3}$

The given quadratic equation is $12x^2-2bx+9=0.$
Comparing this equation with the general quadratic equation,$A{x}^2+Bx+C=0$ we get,
$A=12,B=-2b,C=9$
$\therefore D=B^2-4AC$
$=(-2b{)}^2-4\times 12\times 9$
$=4b^2-432$
For no real roots, $D<0$
$\Rightarrow 4b^2-432<0$
$\Rightarrow 4b^2<4\times 4\times 3\times 3^2$
$\Rightarrow b^2<(6\sqrt{3}{)}^2$
$\Rightarrow -6\sqrt{3}<b<6\sqrt{3}$

Hence, the correct answer is option (3).