Question

The range of $m$ for which the equation $({\log }_{2}x{)}^2-4{\log }_{2}x-m^2-2m-13=0$ has real roots is

• A. $(-\infty ,0)$
• B. $(0,\infty )$
• C. $[1,\infty )$
• D. $(-\infty ,\infty )$

Answer 1

The correct option is D $(-\infty ,\infty )$

$({\log }_{2}x{)}^2-4{\log }_{2}x-m^2-2m-13=0$
Let ${\log }_{2}x=t\Rightarrow x=2^t,t\in R$
$t^2-4t-(m^2+2m+13)=0\cdots \left(1\right)$
For the given equation to have real roots, equation $\left(1\right)$ should have real roots. So,
$D\ge 0\Rightarrow 16+4(m^2+2m+13)\ge 0\Rightarrow m^2+2m+17\ge 0\Rightarrow (m+1{)}^2+16\ge 0\Rightarrow m\in R$