Quadratic Equations for CAT 2026: 19 Formulas + 16 PYQs
Quadratic equations for CAT 2026 are one of the most consistently tested Algebra sub-topics, yet most internet content for this keyword is surprisingly thin. CAT 2026 will not test long-hand factorisation. It will test sum and product of roots, discriminant-based parameter questions, and quadratic combined with inequality. This cheatsheet pins 19 formulas, organised by recognition cue, and ends with 16 CAT-level questions that mirror the patterns setters actually use.
What separates a one-minute solve from a five-minute substitution exercise is knowing which derived identity to reach for. The same question can be solved by long-hand quadratic formula application or by a one-line Vieta substitution. The cheatsheet below is therefore organised by question-pattern recognition, not by alphabetical formula order.
Why Quadratic Equation Content for CAT Is Thinner Than You Think
Search any major SEO ranking page for "quadratic equations CAT" and you find textbook restatements: here is ax squared plus bx plus c, here is the formula, here is one factorisation example. None of that is what CAT actually tests. CAT 2026 will not give you a clean quadratic to factor. It will give you a parameter question where the discriminant decides whether real roots exist. It will give you an expression in roots that needs to be reduced to sum and product. It will combine the quadratic with an inequality and ask for the range of x.
The 19 formulas in this cheatsheet are organised around those question types, not the textbook order. Each formula is tagged with the recognition cue that triggers it, so the leap from question to formula takes seconds, not minutes. This is the gap most internet content misses, and the gap this blog closes for CAT 2026.
The 19 Quadratic Equations Formulas for CAT 2026
The cheatsheet groups 19 formulas into four blocks. Each block has a recognition cue: a sentence describing the kind of question that triggers it. Working block by block embeds the recognition habit that the type-tag drill depends on.
Block 1 — Standard Form and Quadratic Formula (4 formulas)
The standard-form block is the entry point. These four formulas describe the equation, its roots, and the relationship between coefficients. Recognition cue: any direct quadratic equation that needs to be solved or its roots described.
| # | Formula | Use case |
|---|---|---|
| 1 | ax2 + bx + c = 0 (a ≠ 0) | Standard form. |
| 2 | x = [−b ± √(b2 − 4ac)] / 2a | Quadratic formula for roots. |
| 3 | Vertex: x = −b / 2a; y = c − b2/4a | Maximum or minimum of quadratic. |
| 4 | Factored form: a(x − α)(x − β) = 0 | Construct quadratic from roots. |
Block 2 — Sum and Product of Roots (Vieta) (5 formulas)
Vieta's formulas link the roots of a quadratic to its coefficients. This block is the highest-leverage in the topic because most CAT root-expression questions reduce to one or two Vieta substitutions. Recognition cue: the question asks for an expression in both roots without giving the roots explicitly.
| # | Formula | Recognition cue |
|---|---|---|
| 5 | α + β = −b / a | Sum of roots. |
| 6 | αβ = c / a | Product of roots. |
| 7 | α2 + β2 = (α + β)2 − 2αβ | Sum of squared roots. |
| 8 | 1/α + 1/β = (α + β) / (αβ) | Sum of reciprocals of roots. |
| 9 | (α − β)2 = (α + β)2 − 4αβ | Difference of roots squared. |
Block 3 — Discriminant and Nature of Roots (5 formulas)
The discriminant block decides the type of roots without solving. CAT regularly tests these patterns through parameter questions. Recognition cue: the question asks for values of a parameter such that roots are real, equal, distinct, complex, or rational.
| # | Formula / Rule | Use case |
|---|---|---|
| 10 | D = b2 − 4ac | The discriminant. |
| 11 | D > 0: two distinct real roots | Real and distinct. |
| 12 | D = 0: two equal real roots (repeated) | Equal roots case. |
| 13 | D < 0: two complex conjugate roots | No real roots. |
| 14 | D is a perfect square (with rational coefficients): roots are rational | Rational vs irrational roots. |
Block 4 — Quadratic Inequalities and Special Cases (5 formulas)
The inequality block is where CAT setters get most creative. These five rules cover quadratic inequalities, common roots between two quadratics, and the symmetric-function shortcuts that appear in advanced CAT problems. Recognition cue: the question combines a quadratic with an inequality, or involves two quadratics sharing a root.
| # | Formula / Rule | Use case |
|---|---|---|
| 15 | If a > 0, ax2 + bx + c > 0 outside roots, < 0 between roots | Upward-opening inequality. |
| 16 | If a < 0, ax2 + bx + c < 0 outside roots, > 0 between roots | Downward-opening inequality. |
| 17 | Common root of two quadratics: cross-multiplication of coefficients | Two equations share one root. |
| 18 | Quadratic with rational coefficients: irrational roots occur in conjugate pairs (p ± √q) | Conjugate-pair root rule. |
| 19 | Quadratic with real coefficients: complex roots occur in conjugate pairs (p ± iq) | Conjugate complex roots rule. |
Three Quadratic Traps That Recur in CAT Papers
Three traps recur in CAT quadratic questions. The first is forgetting to check the discriminant when a question implicitly assumes real roots. A question that talks about the difference or sum of roots only makes sense if those roots are real, which means the discriminant must be non-negative. Skipping this check gives a wrong answer when the parameter range pushes D below zero.
The second trap is mixing the sign of a in inequality questions. Formulas 15 and 16 are mirror images of each other. Confusing the two flips the direction of the answer interval. The third trap is forgetting that irrational roots of a rational-coefficient quadratic occur in conjugate pairs. CAT setters use this to construct questions where one root is given as p plus root q and ask for the other root, which must be p minus root q.
16 Must-Solve CAT Quadratic Questions
These 16 questions cover all four blocks. Each is tagged with the block and the formula it tests. The drill: solve under timed conditions, target under 90 seconds per question, and use the cheatsheet for recall only after attempting.
Solve 2x2 − 7x + 3 = 0.
D = 49 − 24 = 25. Roots = (7 ± 5)/4 = 3 or 0.5. Answer: x = 3 or 1/2
Form a quadratic with roots 2 and −5.
Sum = −3, product = −10. Equation: x2 + 3x − 10 = 0. Answer: x2 + 3x − 10 = 0
Find the minimum value of x2 − 6x + 11.
Vertex at x = 3. Min value = 9 − 18 + 11 = 2. Answer: 2
If α and β are roots of x2 − 5x + 6 = 0, find α2 + β2.
α + β = 5, αβ = 6. α2 + β2 = 25 − 12 = 13. Answer: 13
For x2 − 8x + 12 = 0, find 1/α + 1/β.
(α + β)/(αβ) = 8/12 = 2/3. Answer: 2/3
For x2 − 7x + 10 = 0, find α − β (taking α > β).
(α − β)2 = 49 − 40 = 9. So α − β = 3. Answer: 3
For what value of k does x2 + kx + 9 = 0 have equal roots?
D = k2 − 36 = 0 gives k = ±6. Answer: k = ±6
For what values of k does x2 + 2x + k = 0 have no real roots?
D = 4 − 4k < 0 gives k > 1. Answer: k > 1
For x2 − 5x + 6 = 0, are the roots rational?
D = 25 − 24 = 1, which is a perfect square. Roots are rational. Answer: Yes (3 and 2)
For what values of k does x2 − 4x + k = 0 have two distinct real roots?
D = 16 − 4k > 0 gives k < 4. Answer: k < 4
Solve x2 − 5x + 6 > 0.
Roots at 2 and 3. Since a > 0, expression is positive outside roots. Answer: x < 2 or x > 3
Solve −x2 + 4x − 3 > 0.
Multiply by −1, flip: x2 − 4x + 3 < 0. Roots 1, 3. Negative between roots. Answer: 1 < x < 3
If one root of a quadratic with rational coefficients is 4 + √7, find the other.
Conjugate pair rule: other root is 4 − √7. Answer: 4 − √7
If α and β are roots of x2 + 3x + 2 = 0, find a quadratic whose roots are 2α and 2β.
New sum = 2(α + β) = −6, new product = 4αβ = 8. Equation: x2 + 6x + 8 = 0. Answer: x2 + 6x + 8 = 0
For how many integer values of k does the equation x2 − (k + 3)x + 4 = 0 have real roots?
D = (k + 3)2 − 16 ≥ 0 gives (k + 3) ≤ −4 or (k + 3) ≥ 4, i.e., k ≤ −7 or k ≥ 1. Infinite integers in both ranges, so the question typically restricts k. If k ∈ {−10, ..., 10}, count: (−10 to −7) = 4 values + (1 to 10) = 10 values = 14 values. Answer: 14 (in [−10, 10])
If α and β are roots of x2 − 4x + 1 = 0, find α3 + β3.
α + β = 4, αβ = 1. α3 + β3 = (α + β)3 − 3αβ(α + β) = 64 − 12 = 52. Answer: 52
Build a 5-Day Quadratic Block Into Your CAT 2026 Plan
Quadratic equations is one of 22 Quants topics that need to fit a CAT 2026 plan. A diagnostic-driven sequence puts quadratics where it earns the most marks per study hour, alongside inequality and logarithm prep.
Sequence My Algebra Cluster PlanWhere Quadratic Equations Fit in the CAT 2026 Algebra Cluster
Quadratics is the connector topic in CAT Algebra. It sits between linear algebra and the higher polynomial work, and it overlaps with inequality and logarithm in roughly half of CAT Algebra questions. A focused 5-day block, scheduled in month two of a 9-month CAT 2026 plan, covers the topic well. Day one to two: Blocks 1 and 2. Day three: Block 3. Day four: Block 4. Day five: 16 mixed questions under timed conditions. By the end of week one, the entire Vieta substitution pattern is automatic.
For a working professional or college student, this 5-day block can stretch to 10 days with shorter study windows, which still puts the topic comfortably within month two. For sequencing the rest of the Algebra cluster, our CAT exam guide walks through the topic priorities, and the CAT 2026 waitlist details page explains how the personalised planner builds your topic sequence.
Three Reflexes That Compress Quadratic Solves to Under 60 Seconds
Once 19 formulas are memorised, three reflexes separate aspirants who finish quadratic questions in 60 seconds from those who take three minutes. Reflex one: Vieta first. Before solving for roots, check if the question can be answered using sum and product alone. About sixty percent of CAT root-expression questions can. Reflex two: discriminant check. For any parameter question, write D explicitly before solving for the parameter. Reflex three: sign analysis. For quadratic inequality questions, write the sign of a explicitly, then the roots, then the sign in each interval. These three install through timed drill. The CAT practice questions library on Optima Learn tags quadratic problems by block so each reflex can be drilled separately, and the CAT preparation blogs library has companion cheatsheets on Logarithms, AP GP, and Number System.
Common Doubts About Quadratic Equations Preparation for CAT 2026
How many quadratic questions appear in CAT 2026?
Expect 1 to 2 direct quadratic questions plus 2 to 3 adjacent Algebra and Inequality questions that use quadratic identities. That makes the topic a 3 to 4 mark contributor on average. Given that 19 formulas cover the topic, this is a strong return on the focused 5-day preparation block.
Is Vieta's formulas worth memorising over factorisation?
Yes, by a large margin. Factorisation works on textbook-clean quadratics but fails when CAT setters use awkward coefficients. Vieta's formulas always work, are faster to apply, and reduce most CAT root-expression questions to one or two substitutions. Memorising Vieta cold is the single highest-leverage move in this topic.
Should I memorise the formulas for higher-power root expressions?
Memorise three: alpha squared plus beta squared, alpha cubed plus beta cubed, and one over alpha plus one over beta. Higher powers are derived on the fly using power sums. Memorising more than these three costs time without a proportional return on CAT questions.
How do I revise quadratic equations one week before CAT 2026?
A one-week revision plan: day one, re-read 19 formulas. Day two and three, drill Vieta and discriminant with five questions each. Day four, drill inequality with five questions. Day five, attempt 16 mixed-block questions under timed conditions. Day six, review every error. Day seven, scan the cheatsheet for 15 minutes only before the exam.
Final note. Quadratic equations for CAT 2026 reduce to 19 formulas across four blocks, with Vieta's formulas carrying the most weight. The topic rewards recognition over factorisation, which is what most internet content gets wrong. Drill block-by-block, build the three reflexes, and the CAT score predictor alongside mocks will track the improvement.
