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JEE Mains Maths · Quadratic Equations

Roots and Coefficients

Reading the sum and product of the roots straight from the coefficients, and turning any condition on the roots into equations in those two numbers.

All Quadratic Equations notesJEE Mains Maths strategy

Why this matters

Sixteen PYQs, fourteen of them multiple choice, and eight from 2026. Twelve turn a condition on two roots into equations in their sum and product — a fixed difference, one root twice the other, roots in G.P., or a root known in advance; four go past degree two, to Vieta for a cubic, a quadratic divisor, or a cubic fixed by four values. Two ideas cover the page.

Concept 1 of 2: Sum, product and difference of roots

The roots themselves are rarely needed. Their sum −ba-\frac ba and product ca\frac ca come straight from the coefficients, and any condition that treats the two roots alike — their difference, their ratio, their reciprocals — can be written in those two numbers. The difference comes through its square: (β−α)2=(α+β)2−4αβ(\beta-\alpha)^2=(\alpha+\beta)^2-4\alpha\beta, which is the discriminant over a2a^2. If one root is known, the other is the product divided by it.

Definition

  • α+β=−ba\alpha+\beta=-\frac{b}{a}, αβ=ca\alpha\beta=\frac{c}{a}.
  • (β−α)2=(α+β)2−4αβ=Da2(\beta-\alpha)^2=(\alpha+\beta)^2-4\alpha\beta=\frac{D}{a^2}.
  • a+b+c=0a+b+c=0: one root is 11, the other ca\frac{c}{a}.
  • Roots tt and ktkt: eliminate tt between the sum and the product.

Sum, product, difference

α+β=−ba,αβ=ca,(β−α)2=b2−4aca2\alpha+\beta=-\frac{b}{a},\quad \alpha\beta=\frac{c}{a},\quad (\beta-\alpha)^2=\frac{b^2-4ac}{a^2}

Worked example

One root of 2x2−9x+k=02x^2-9x+k=0 is twice the other. Find kk.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 22 Jan 2026 Shift 2 · Q63Moderate

Example 1 · Quadratic Equations · Roots and Coefficients

Let α,β\alpha,\beta be the roots of the quadratic equation 12x2−20x+3λ=0,λ∈Z12x^{2}- 20x + 3\lambda = 0,\lambda \in Z. If 12≤∣β−α∣≤32\frac{1}{2}\leq |\beta - \alpha| \leq\frac{3}{2}, then the sum of all possible values of λ\lambda is :

The difference needs its square

β−α\beta-\alpha changes sign when the roots swap, so it cannot be written in the sum and product directly. Work with (β−α)2(\beta-\alpha)^2, and take the square root only at the end.
Drill 11 more on sum, product and difference of roots

Concept 2 of 2: Beyond degree two: cubics and polynomial divisors

The same idea runs for a cubic ax3+bx2+cx+dax^3+bx^2+cx+d: the coefficients give the sum of the roots, the sum of their products in pairs, and their product. A missing x2x^2 term means the roots add to 00. For divisibility by a quadratic, the roots of the divisor must also be roots of the cubic — substitute them, or divide and set every coefficient of the remainder to 00.

Definition

  • Cubic: α+β+γ=−ba\alpha+\beta+\gamma=-\frac{b}{a}, αβ+βγ+γα=ca\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}, αβγ=−da\alpha\beta\gamma=-\frac{d}{a}.
  • p(x)p(x) divisible by q(x)q(x): the remainder is 00 for every xx, so each of its coefficients is 00.
  • A polynomial fixed by its values at several points: build a helper that is 00 at those points, and factor it.

Vieta for a cubic

α+β+γ=−ba,∑αβ=ca,αβγ=−da\alpha+\beta+\gamma=-\frac{b}{a},\quad \sum\alpha\beta=\frac{c}{a},\quad \alpha\beta\gamma=-\frac{d}{a}

Worked example

For which p,qp,q is x3+px2+qx+6x^3+px^2+qx+6 divisible by x2−1x^2-1?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 28 Jan 2026 Shift 1 · Q58Moderate

Example 2 · Quadratic Equations · Roots and Coefficients

Let S={x3+ax2+bx+c:a,b,c∈NS=\left\{ x^{3}+ax^{2}+bx+c:a,b,c\in N \right. and a,b,c≤20}a,b,c\leq 20\} be a set of polynomials. Then the number of polynomials in S , which are divisible by x2+2x^{2}+ 2, is

The signs alternate

For a cubic the sum of the roots is −ba-\frac{b}{a}, the pair sum +ca+\frac{c}{a}, and the product −da-\frac{d}{a} — not da\frac{d}{a}. The signs go minus, plus, minus.
Drill 3 more on beyond degree two: cubics and polynomial divisors

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Sum, product and difference of roots

    Sum, product, difference

    α+β=−ba,αβ=ca,(β−α)2=b2−4aca2\alpha+\beta=-\frac{b}{a},\quad \alpha\beta=\frac{c}{a},\quad (\beta-\alpha)^2=\frac{b^2-4ac}{a^2}
  • Beyond degree two: cubics and polynomial divisors

    Vieta for a cubic

    α+β+γ=−ba,∑αβ=ca,αβγ=−da\alpha+\beta+\gamma=-\frac{b}{a},\quad \sum\alpha\beta=\frac{c}{a},\quad \alpha\beta\gamma=-\frac{d}{a}

Watch out for (2)

Test yourself on Quadratic Equations

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Quadratic Equations chapter testDrill all 16 questions