Question

If the roots of the equation $x^2+2(m-1)x+(m+5)=0$ are real and equal, find the value of $m$.

Hint:

"Real and equal roots" \Rightarrow\ set the discriminant to zero. Always simplify first (factor out common constants) to avoid large numbers.

Answer 1

For real and equal roots, discriminant $D=0$. Here $a=1,\ b=2(m-1),\ c=m+5$.
\[ D=b^2-4ac=[2(m-1)]^2-4(m+5)=0 \] \[ \Rightarrow\ 4(m-1)^2-4(m+5)=0 $\Rightarrow$ (m-1)^2-(m+5)=0 \] \[ \Rightarrow\ m^2-2m+1-m-5=0 $\Rightarrow$ m^2-3m-4=0 \] \[ \Rightarrow\ (m-4)(m+1)=0 \Rightarrow\ m=4\ \text{or}\ m=-1. \] \boxed{m=4\ \text{or}\ m=-1}