Q.Without solving, determine the number of real roots of $x^2 + 2x + 5 = 0$ using the discriminant, and also verify whether $x = -1$ is a root.

Two independent checks here: the discriminant's sign tells us the root count, and direct substitution checks the specific candidate.

Part (i) — discriminant: here $a=1, b=2, c=5$. $D=b^2-4ac=(2)^2-4(1)(5)=4-20=-16$. Since $D<0$, the equation has no real root.

Part (ii) — verify $x=-1$: substitute into $p(x)=x^2+2x+5$: $p(-1)=(-1)^2+2(-1)+5=1-2+5=4$. Since $p(-1)=4\neq0$, $x=-1$ is not a root. …

A subtle mistake is treating the discriminant check and the substitution check as unrelated facts that could disagree — since $D<0$ means NO real number can satisfy the equation, a substitution check on ANY real candidate value must always come out non-zero; if a subs …