Verifying Whether a Given Number is a Root

Sometimes a problem gives a candidate value and asks only whether it is a root of a given quadratic equation, without asking to solve the equation from scratch. This is answered directly from the definition of a root in §2: substitute the candidate value for $x$ into $p(x) = ax^2+bx+c$ and evaluate.

Method:

1. Substitute the given number, say $x=k$, into $p(x) = ax^2+bx+c$.
2. Simplify the resulting numerical expression.
3. If $p(k) = 0$, then $k$ is a root of the equation. If $p(k) \neq 0$, then $k$ is not a root.

This check is a direct, one-line application of the definition — it never requires factorizing, completing the square, or using Sridharacharya's formula, and it works even for equations that would otherwise be tedious to solve fully.