PYQ Vault

Playbook

Quadratic Equations

Vieta and the discriminant, plus equations that are quadratics after a substitution. Most wrong answers keep a root the substitution should have dropped.

Questions in the bank
119
q/paper in 2025–26
1.00
Numeric answer
32%
Notes pages
6

Tier: Core

When you’ll see it

Roots α and β of a quadratic, a condition on those roots, or an equation that becomes a quadratic after a substitution.

How this chapter is tested

The chapter is built on Vieta: α + β = −b/a and αβ = c/a. Most conditions on the roots — a fixed difference, one root twice the other, α² + β², even α²⁵ + β²⁵ — become equations in the sum and product. High powers use the recurrence the equation itself gives: aPₙ + bPₙ₋₁ + cPₙ₋₂ = 0, where Pₙ = αⁿ + βⁿ.

The other half is equations that are quadratic in disguise. Exponential equations take t = aˣ, logarithmic ones take logs or change the base, a repeated block such as x + 1/x becomes the unknown, and moduli split the number line at their critical points. Each substitution limits which roots count: aˣ is positive, a log needs a valid base, |x| is never negative. Most wrong answers keep a root that should have been dropped.

The discriminant page decides real, equal or rational roots, and places the roots against a number using the discriminant, the sign of f(k) and the vertex. Questions that ask for the number of real roots, or the sum of all of them, reward a careful count more than hard algebra. The chapter feeds Complex Numbers, and Probability when coefficients are rolled on dice.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Roots and coefficients

    α + β = −b/a, αβ = c/a; for a cubic the signs of sum, pair sum and product go minus, plus, minus.

  • Symmetric functions and power sums

    α² + β² = (α + β)² − 2αβ; high powers by aPₙ + bPₙ₋₁ + cPₙ₋₂ = 0.

  • Common roots and new equations

    Eliminate the x² term to find a shared root; an equation with given roots is x² − (sum)x + (product) = 0.

  • Discriminant and location of roots

    With real coefficients, D > 0, D = 0, D < 0 give distinct real, equal and non-real roots; place roots with D, a·f(k) and the vertex −b/2a.

  • Modulus and greatest integer

    Split at the critical points, or take |x| as the unknown when the equation is even in x; {x} lies in [0, 1).

  • Equations reducible to quadratics

    t = aˣ, a log, or a repeated block as the unknown; check each t against the values its block can take.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • The sum is −b/a, not −b

    Read the sum and product only after dividing by the leading coefficient.

  • A root the substitution cannot reach

    t = aˣ must be positive and t = |x| cannot be negative; such roots give no x. And t = 0 for |x| gives one root, not two.

  • The leading coefficient can vanish

    When a holds a parameter, the value that makes it 0 leaves a linear equation, where the discriminant test does not apply.

  • The vertex condition left out

    D ≥ 0 and a·f(k) > 0 hold both when both roots exceed k and when both are below it. The vertex −b/2a decides which.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Quadratic Equations notes

Drill every Quadratic Equations question

119 questions from the bank, across 6 subtopics.

Drill the 119 questions

Drill one subtopic at a time

The 6 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.