Question

If $x^2+2(a-1)x+a+5=0$ has real roots to the interval $(1,3)$, then complete set of value of ${}^{\prime }{a}^{\prime }$ is

• A. $(-\infty ,-\frac{8}{7})$
• B. $(4,\infty )$
• C. $(-\infty ,-\frac{48}{3})$
• D. $aϵ(-\frac{8}{7},-1]$

Answer 1

The correct option is D $aϵ(-\frac{8}{7},-1]$

Given $f\left(x\right)=x^2+2(a-1)x+a+5=0$ has real roots $⟹b^2-4ac\ge 0$ in $a{x}^2+bx+c=0$

$⟹(a-1{)}^2-(a+5)\ge 0⟹a\in (-\infty ,-1]\cup [4,\infty )\cdots \cdots (1)$

the roots lies in $(1,3)⟹f\left(1\right).f\left(3\right)>0⟹(3a+4)(7a+8)>0$

$⟹a\in (-\infty ,-\frac{4}{3})\cup (-\frac{8}{7},\infty )\cdots \cdots \left(2\right)$

from $\left(1\right),\left(2\right)$ $a\in (-\frac{8}{7},-1]$