मूल ज्ञात करने का चिह्न-नियम शॉर्टकट
The Sign Rule Shortcut for Finding Roots
Once factorization is comfortable, a sign-rule shortcut lets you read the nature of the roots almost instantly from the coefficients, which is invaluable in the comparison format. Instead of fully solving, you often only need to know whether each root is positive or negative, and the signs of b and c hand you that information directly.
The Four Sign Cases
Look only at the signs of the middle term and the constant, then read the root signs from the table below. This works because the sum and product of the roots are tied to those two coefficients. Memorising these four cases means you can predict the sign pattern of the roots before you have even found their values.
| Middle (b) | Constant (c) | Both roots |
|---|---|---|
| + | + | both negative |
| − | + | both positive |
| + | − | one +, one − (larger negative) |
| − | − | one +, one − (larger positive) |
Why This Saves Time in Comparison
In the x-versus-y format you frequently do not need exact values, only whether every root of one equation beats every root of the other. If one equation gives two positive roots and the other gives two negative roots, you can declare the answer without finishing the arithmetic. The sign rule is what surfaces these shortcuts, turning a two-minute pair of solves into a fifteen-second decision.
Using the Discriminant Sparingly
The discriminant b² − 4ac tells you whether roots are real and distinct, equal, or imaginary, but bank quadratics are built to have clean real roots, so you rarely need it. Keep it in reserve only for the odd question where a split refuses to appear; even then, a quick check usually reveals a misread sign rather than genuinely irrational roots, so recheck your split before you trust the discriminant.
Solved Example
Example: x² − 9x + 18 = 0. Middle −, constant +, so both roots are positive; splitting confirms 3 and 6. If the paired y-equation had both-negative roots, x > y follows instantly.
✗ Solving both equations fully before comparing | ✓ Reading root signs first and stopping early when one clearly dominates
A Practice Pattern for the Sign Rule
The sign rule only becomes fast if you drill it against the comparison, not in isolation. A good practice habit is to look at a pair of equations and, before solving either, write down the sign pattern each one must produce. If x's equation reads (−, +) you note 'both positive' and if y's reads (+, +) you note 'both negative', and you can already see x > y without a single multiplication. When both equations share the same sign pattern, you know in advance that you will need actual values and a number line, so you budget your time accordingly. Training the eye to read sign patterns first is what separates a twenty-second solver from a two-minute one on this block.
मूलों के चिह्न b और c के चिह्न से पढ़ें: (+,+) दोनों ऋणात्मक; (−,+) दोनों धनात्मक; c ऋणात्मक हो तो एक धन एक ऋण। इससे तुलना बहुत तेज़ होती है।
Exam Pointer — The sign rule is the single biggest time-saver in the SBI Clerk quadratic block. Many comparison questions can be answered from root signs alone, without exact values. Train yourself to check signs first and only compute full roots when the sign pattern leaves the comparison genuinely undecided.
60-Second Recap
- Root signs follow the signs of b and c — memorise the four cases.
- (+,+) → both negative; (−,+) → both positive.
- c negative → one positive, one negative root.
- In comparison questions, signs often decide the answer alone.