CAT Quant · Study & Practice

Quadratic Equations

AreaAlgebra DifficultyModerate CAT weightage1–3 questions (directly + inside maxima-minima, functions, progressions, geometry word problems)

A quadratic is any equation that can be pushed into the form ax² + bx + c = 0 with a ≠ 0, and almost nothing in CAT Algebra works without it. The moment a word problem multiplies two unknowns — area, the product of two consecutive numbers, distance at two speeds, a profit that depends on price — a quadratic is hiding underneath. CAT very rarely asks you to merely "solve x² − 5x + 6 = 0". Instead it tests whether you can read the discriminant to decide how many real roots exist, build an equation backwards from a stated sum and product, or find the largest value an expression can reach without calculus. The good news is that one compact toolkit covers all of it: the quadratic formula, D = b² − 4ac for the nature of the roots, sum = −b/a and product = c/a (Vieta), and the vertex result that ax² + bx + c hits its extreme value −D/4a at x = −b/2a. This chapter drills each of these with worked CAT-style examples, the fastest exam method, and the traps — sign errors, a forgotten "a ≠ 0", and treating a downward parabola as if it had a minimum — that quietly cost marks.

Topics

1 Roots & Nature of Roots 4 worked · 5 practice · 8-Q test Start → 2 Factorization 4 worked · 5 practice · 8-Q test Start → 3 Sum & Product of Roots 4 worked · 5 practice · 8-Q test Start → 4 Maximum & Minimum 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Read D = b² − 4ac first: D > 0 two real roots, D = 0 equal roots, D < 0 none. "Equal roots" and "real roots" are pure D conditions — never solve the full formula.
  • If a, b, c are rational and D is a perfect square, the roots are rational ⇒ factorisation works. If not, jump straight to the formula.
  • Vieta beats solving: α + β = −b/a, αβ = c/a. Get α² + β² from (α + β)² − 2αβ and (α − β)² from D/a² without finding the roots.
  • Build an equation from roots instantly: x² − (sum)x + (product) = 0. For transformed roots, recompute the new sum and product, then rebuild.
  • Optimisation without calculus: vertex at x = −b/2a, extreme value −D/4a. a > 0 ⇒ minimum, a < 0 ⇒ maximum.
  • Fixed sum S ⇒ product is largest when the parts are equal (S²/4); fixed product ⇒ sum is least when equal. A square maximises area for a fixed perimeter.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Dropping the a in α + β = −b/a or αβ = c/a when a ≠ 1 (e.g. for 2x² − 5x + 3, the sum is 5/2, not 5).
  • Reporting a minimum for a downward parabola (a < 0) — it has a maximum and no minimum.
  • Sign errors in the discriminant: writing b² + 4ac, or mishandling a negative b so that −b/2a comes out wrong.
  • Forgetting the a ≠ 0 condition — if the leading coefficient can be zero the equation may be linear, with only one root.
  • Mixing up the equation-building signs: it is x² − (sum)x + (product), so a negative sum makes the middle term positive.

📈 CAT exam insight & PYQ analysis

In CAT, pure quadratics are uncommon as standalone questions; they surface most often through the discriminant (nature-of-roots and "find k" conditions) and through maxima-minima dressed up as area, product, or expense optimisation. XAT and SNAP are more willing to ask direct root, sum-product, and equation-building questions. The recurring high-value patterns are: equal/real-root conditions on a parameter, building a new equation from transformed roots, and finding a maximum or minimum value via the vertex. Difficulty is Moderate, and the questions that trip people are sign-sensitive — a single slip in b² − 4ac or −b/2a flips the answer. Prioritise the discriminant and the vertex shortcut for the best return.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Quadratic formula?Tap to reveal
x = [−b ± √(b² − 4ac)] / 2a
Discriminant D = ?Tap to reveal
b² − 4ac
D > 0 means?Tap to reveal
Two distinct real roots
D = 0 means?Tap to reveal
Real and equal roots, x = −b/2a
D < 0 means?Tap to reveal
No real roots (complex conjugate pair)
Sum of roots?Tap to reveal
−b/a
Product of roots?Tap to reveal
c/a
α² + β² in terms of sum and product?Tap to reveal
(α + β)² − 2αβ
Equation from roots α, β?Tap to reveal
x² − (α + β)x + αβ = 0
Vertex x-coordinate?Tap to reveal
−b/2a
Extreme value of ax² + bx + c?Tap to reveal
−D/4a (min if a > 0, max if a < 0)
Two numbers with fixed sum S have max product?Tap to reveal
S²/4 (when both equal S/2)

📌 Quick revision

A quadratic is ax² + bx + c = 0 (a ≠ 0), solved by x = [−b ± √(b² − 4ac)]/2a. The discriminant D = b² − 4ac fixes the nature of the roots: D > 0 two real, D = 0 equal, D < 0 none. Factorise when D is a perfect square; otherwise use the formula. By Vieta, α + β = −b/a and αβ = c/a, and any symmetric expression in the roots follows from these. Build an equation from roots with x² − (sum)x + (product) = 0. For optimisation, the vertex sits at x = −b/2a with extreme value −D/4a — a minimum if a > 0, a maximum if a < 0; a fixed sum gives the largest product when the parts are equal.

Chapter test

📝 Quadratic Equations — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
📊 My CAT analytics
Track scores, accuracy by topic and progress over time.
View analytics →

🏆 Vidaara CAT success checklist

You have truly mastered Quadratic Equations when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards