Parameter-Based Quadratic Tricks in 59 Seconds! 🔥 | JEE Maths

 

❓ Concept

Parameter-Based Quadratic Tricks (α, β) in 59 Sec

Whenever you see a quadratic with p, k, λ in coefficients, panic mat karo.
👉 Parameter-based quadratics follow a fixed golden framework — once you apply it step-by-step, answers come automatically.

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1️⃣ Quadratic with Parameter

General form:

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

Here,

$$a,\ b,\ c \quad \text{depend on a parameter } p$$

Roots are:

$$\alpha,\ \beta$$

👉 Goal in JEE questions:
Find range of p such that roots satisfy a given condition.

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2️⃣ Real Roots Condition (ALWAYS STEP 1)

For real roots:

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

📌 This step is non-negotiable.
Most students forget it and lose marks.

👉 Solve:

$$b^2 - 4ac \ge 0 \Rightarrow \text{range of } p$$

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3️⃣ Sign of Roots (α, β)

Use standard relations:

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

✔ For both roots positive

$$\alpha + \beta > 0$$
$$\alpha\beta > 0$$

✔ For both roots negative

$$\alpha + \beta < 0$$
$$\alpha\beta > 0$$

👉 These conditions give extra inequalities in p.

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4️⃣ Roots in an Interval (L, R)

If question says:

Both roots lie in $(L, R)$

Then apply interval test:

For $a > 0$:

$$f(L) > 0$$
$$f(R) > 0$$

For $a < 0$:

$$f(L) < 0$$
$$f(R) < 0$$

👉 Plug L and R directly into the quadratic.
Each gives an inequality in p.

📌 This trick avoids solving roots completely.

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5️⃣ Golden Framework (FINAL CHECKLIST 🔥)

🟡 Step 1:

$$D \ge 0$$

🟡 Step 2:
Sign of $\alpha + \beta$

🟡 Step 3:
Sign of $\alpha\beta$

🟡 Step 4:
Interval test using $f(L)$, $f(R)$

🟡 Step 5:
👉 Intersect all p-ranges

Final answer = common range of p

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✅ Final Takeaway

🔥 Parameter Quadratic Master Rule:

• Never find roots directly

• Discriminant first

• Then sum & product

• Interval test last

• Combine everything

This single framework solves 90% of JEE parameter questions.