Question

The equation $x^2+2(m-1)+m+5=0$ has at least one positive root. Determine the range for $m$.

Answer 1

Given,

$x^2+2(m-1)x+m+5=0$

Condition for quadratic equation to have at least one positive root is:

$b^2\ge 4ac$

$\left[2\right(m-1){]}^2\ge 4(1\left)\right(m+5)$

$4(m-1{)}^2\ge 4(m+5)$

$m^2+1-2m\ge m+5$

$m^2-3m-4\ge 0$

$(m+1)(m-4)\ge 0$

$\therefore -1\le m\le 4$ is the range.