JEE Main Shortcut: Kab Quadratic ke Dono Roots Negative Hote Hain? 💡

 

❓ Concept

Both Roots Negative Trick in 59 Sec

Finding when both roots of a quadratic equation are negative is one of the most scoring and repetitive JEE topics.
Good news? 👉 It follows just 3 golden rules.

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1️⃣ Quadratic Form

General quadratic:

$$ax^2 + bx + c = 0$$

First, ensure:

$$a > 0$$

📌 If $a < 0$, multiply the equation by –1 before applying the rules.

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2️⃣ Condition 1 — Roots Must Be Real

For real roots:

$$D = b^2 - 4ac \ge 0$$

This condition is mandatory.
No real roots → no sign discussion.

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3️⃣ Condition 2 — Sum of Roots

$$\alpha + \beta = -\frac{b}{a}$$

For both roots negative:

$$\alpha + \beta < 0$$

Since $a > 0$:

$$-\frac{b}{a} < 0 \Rightarrow b > 0$$

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4️⃣ Condition 3 — Product of Roots

$$\alpha\beta = \frac{c}{a}$$

For both roots negative:

$$\alpha\beta > 0$$

Again, since $a > 0$:

$$c > 0$$

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5️⃣ Final 3 Golden Rules 🔥

If:

$$\boxed{ \begin{aligned} &b^2 - 4ac \ge 0 \\ &b > 0 \\ &c > 0 \end{aligned}}$$

👉 Both roots are negative.

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✅ Final Takeaway

🧠 Super Shortcut:

• Check discriminant first

• Then check sign of $b$

• Then check sign of $c$

No root calculation needed — 100% logic-based.