Quadratic equations are one of the most scoring topics inside CAT Algebra — yet most aspirants lose easy marks here because they rush the factorisation step or forget the sum-and-product shortcut. This guide fixes that. You get 10 verified CAT previous year questions (1997–2025) with year and slot mentioned, followed by 40 CAT-pattern practice questions, all solved step-by-step — organised the way an actual CAT Quant paper is structured, along with the formulas and shortcuts examiners test.
Quick answer: A quadratic equation is any equation of the form ax² + bx + c = 0 (a ≠ 0). In CAT, it usually shows up as direct root-finding, sum/product-of-roots questions, nature-of-roots (discriminant) questions, or as a hidden step inside Number Systems, Inequalities and Functions questions. Based on the confirmed CAT papers referenced in this article, quadratic equations have appeared in almost every CAT year from 1997 to 2025 — sometimes directly, sometimes as the final step of a Functions or Number Systems question.
A note on how this article was built: The 10 previous-year questions were cross-checked with multiple CAT-preparation sources and re-solved to verify their year, slot, and answers. They are presented in original wording while preserving the mathematical conditions and answers. The remaining 40 questions are original CAT-pattern practice questions and are clearly labelled as such.
Why Quadratic Equations Matter in CAT Quant
Algebra is the single largest scoring area inside CAT Quantitative Ability, and quadratic equations form its foundation. Even when a question is officially tagged under Functions, Inequalities or Number Systems, solving it often collapses down to a quadratic equation in the last step.
| Quant Topic Area | Approx. Weightage (Questions) | Where Quadratic Equations Help |
|---|---|---|
| Algebra (overall) | 6–8 out of 22 QA questions | Core topic |
| Quadratic Equations (direct) | 0–2 questions | Direct application |
| Number Systems | 3–5 questions | Integer-root and factorisation logic |
| Inequalities & Modulus | 2–3 questions | Sign-scheme and interval logic |
| Functions & Graphs | 1–2 questions | Roots as x-intercepts |
As per the official CAT exam pattern on iimcat.ac.in, the Quantitative Ability section carries 22 of the exam’s 68 total questions, to be solved in 40 minutes, with +3 marks for every correct answer and −1 for a wrong MCQ response (no negative marking on TITA questions). That timing pressure is exactly why speed-based quadratic equation tricks matter more than textbook methods on exam day.
CAT Quadratic Equations Formulas You Must Know
Before attempting CAT quadratic equations questions with solutions, lock these formulas into memory. Almost every question in this article — and in the actual CAT paper — is built on one of these.
| Concept | Formula |
|---|---|
| Standard form | ax² + bx + c = 0, where a ≠ 0 |
| Roots (Quadratic formula) | x = [−b ± √(b² − 4ac)] / 2a |
| Discriminant | D = b² − 4ac |
| Nature of roots — real & distinct | D > 0 |
| Nature of roots — real & equal | D = 0 |
| Nature of roots — no real roots (complex) | D < 0 |
| Sum of roots | α + β = −b/a |
| Product of roots | αβ = c/a |
| Equation from given roots | x² − (α + β)x + αβ = 0 |
| α² + β² | (α + β)² − 2αβ |
| α³ + β³ | (α + β)³ − 3αβ(α + β) |
| |α − β| | √[(α + β)² − 4αβ] |
| Roots in ratio p : q | b²/ac = (p + q)²/pq |
Top Tricks & Shortcuts for CAT Quadratic Equations
Use these CAT quadratic equations tricks to shave seconds off every question — critical when you have roughly 109 seconds per QA question.
- Split the middle term by inspection first. Check if b² − 4ac is a perfect square before reaching for the formula — if it is, the equation factorises cleanly.
- Use sum-and-product instead of solving for roots. Most CAT questions ask for α² + β², α³ + β³, or 1/α + 1/β — never solve for the actual roots when a direct identity exists.
- Substitute to reduce higher-degree or exponential equations. Expressions like x⁴ − 13x² + 36 = 0, (x + 1/x)² − 3(x + 1/x) + 2 = 0, or 9^t − 4·3^(t+1) + 27 = 0 all become simple quadratics the moment you substitute a single variable for the repeating expression.
- Remember how aᵇ = 1 splits into three cases. This is the exact logic behind CAT 2020’s “distinct positive integer solutions” question: aᵇ = 1 when a = 1, or when b = 0 (a ≠ 0), or when a = −1 and b is even. Missing the third case is the single most common mistake on this question type.
- Sign-scheme for inequalities. For ax² + bx + c > 0 or < 0, plot the roots on a number line and use the “positive-outside, negative-inside” rule (for a > 0) instead of testing multiple values.
- Watch for extraneous roots. Any time you square both sides (as in radical equations), always verify the final answer in the original equation.
- Remember the “sum of roots = 0” shortcut. If a question says roots are “equal in magnitude but opposite in sign,” you only need the coefficient of x to be zero — no discriminant work needed.
Quick Answer Key — All 50 Questions
Use this table for a fast self-check. Q1–Q10 are verified CAT PYQs; Q11–Q50 are practice questions.
| Q. No. | Answer | Q. No. | Answer | Q. No. | Answer |
|---|---|---|---|---|---|
| 1 | c = −15 | 18 | Sum = 7, Product = 10 | 35 | 8 and 11 |
| 2 | Roots = 6, 1 | 19 | 29/4 | 36 | x = 5 or 1/5 |
| 3 | Min value = 5 | 20 | 3/4 | 37 | 40 km/h |
| 4 | n = 24 | 21 | 52 | 38 | x = ±2, ±3 |
| 5 | Min value = 3 | 22 | k = 7, other root = 4 | 39 | x = −1, 1, 2, 4 |
| 6 | Product = −16 | 23 | k = 8 | 40 | x = 0, 1 |
| 7 | b² + c = 549 | 24 | Real & equal | 41 | x = 9 (x = 2 rejected) |
| 8 | 6 solutions | 25 | No real roots | 42 | x = 1, 4 |
| 9 | 1 distinct real root | 26 | Real, rational, distinct | 43 | 2 < x < 3 |
| 10 | Product = 20 | 27 | k = ±6 | 44 | x ≤ −2 or x ≥ 3 |
| 11 | x = 2, 3 | 28 | k ≤ 1 (k ≠ 0) | 45 | x = 0, 2, 4, 6 |
| 12 | x = 3, 4 | 29 | m = −5, 3 | 46 | 6b² = 25ac |
| 13 | x = −1/2, 2 | 30 | x² − 3x + 2 = 0 | 47 | a = −1 |
| 14 | x = −4, 3 | 31 | x² − 13x + 36 = 0 | 48 | k = 1, 4 |
| 15 | x = 2/3, 1 | 32 | x² − 7x + 1 = 0 | 49 | 2 real roots |
| 16 | x = 4, −2 | 33 | x² − 7x + 12 = 0 | 50 | 4 integer solutions |
| 17 | x = 1/2, −1/3 | 34 | 10 and 11 |
Part 1: Verified CAT Previous Year Questions (Q1–Q10)
These 10 questions are confirmed CAT PYQs, each cross-verified for its year and slot. The problem statements are written in original wording; the numbers, conditions and final answers match the actual paper exactly.
Q1. [CAT 1997] The roots x₁ and x₂ of the equation x² − 2x + c = 0 also satisfy the relation 7x₂ − 4x₁ = 47. Find the value of c.
Solution: From the equation, x₁ + x₂ = 2 (sum of roots), so x₁ = 2 − x₂. Substituting into 7x₂ − 4x₁ = 47: 7x₂ − 4(2 − x₂) = 47 → 7x₂ − 8 + 4x₂ = 47 → 11x₂ = 55 → x₂ = 5, so x₁ = −3. Since c is the product of the roots (c/a with a = 1): c = x₁ · x₂ = (−3)(5) = −15.
Answer: c = −15
Q2. [CAT 2001] Two students attempted to solve the same quadratic equation. The first student copied the coefficient of x correctly but made an error in the constant term, and ended up with roots 4 and 3. The second student copied the constant term correctly but made an error in the coefficient of x, and ended up with roots 3 and 2. Find the correct roots of the original equation.
Solution: The first student’s equation (roots 4, 3) is x² − 7x + 12 = 0 — since he made the error only in the constant term, his coefficient of x (i.e., −7) is correct. The second student’s equation (roots 3, 2) is x² − 5x + 6 = 0 — since he made the error only in the coefficient of x, his constant term (i.e., 6) is correct. Combining the correct coefficient of x (−7) with the correct constant (6), the original equation is x² − 7x + 6 = 0 → (x − 6)(x − 1) = 0. Answer: Roots = 6, 1
Q3. [CAT 2003] Let p and q be the roots of x² − (k − 2)x − (k + 1) = 0, where k is a real parameter. Find the minimum possible value of p² + q².
Solution: Sum of roots: p + q = k − 2. Product of roots: pq = −(k + 1). Using p² + q² = (p+q)² − 2pq: p² + q² = (k−2)² − 2(−(k+1)) = (k−2)² + 2k + 2 = k² − 4k + 4 + 2k + 2 = k² − 2k + 6. This is itself a quadratic in k that opens upward, so its minimum occurs at k = −(−2)/(2×1) = 1. Substituting k = 1: 1 − 2 + 6 = 5.
Answer: Minimum value = 5
Q4. [CAT 2017, Slot 1] If f(x) = x² + 11x + n and g(x) = x, find the largest positive integer value of n for which f(x) = g(x) has two distinct real roots.
Solution: Setting f(x) = g(x): x² + 11x + n = x → x² + 10x + n = 0. For two distinct real roots, the discriminant must be positive: 10² − 4(1)(n) > 0 → 100 − 4n > 0 → n < 25. The largest positive integer satisfying this is n = 24.
Answer: n = 24
Q5. [CAT 2017, Slot 2] Find the minimum possible value of the sum of squares of the roots of the equation x² + (a + 3)x − (a + 5) = 0, where a is real.
Solution: Sum of roots = −(a + 3), product of roots = −(a + 5). Sum of squares = (sum)² − 2(product) = (a+3)² − 2(−(a+5)) = (a+3)² + 2a + 10 = a² + 6a + 9 + 2a + 10 = a² + 8a + 19. Completing the square: a² + 8a + 19 = (a + 4)² + 3. The minimum value of (a+4)² is 0 (at a = −4), so the minimum of the whole expression is 3.
Answer: Minimum value = 3
Q6. [CAT 2019, Slot 1] Find the product of the distinct roots of |x² − x − 6| = x + 2.
Solution: Since the right-hand side must be non-negative, we need x ≥ −2. Case 1: x² − x − 6 = x + 2 → x² − 2x − 8 = 0 → (x−4)(x+2) = 0 → x = 4 or x = −2 (both satisfy x ≥ −2). Case 2: −(x² − x − 6) = x + 2 → x² − 4 = 0 → x = 2 or x = −2 (both satisfy x ≥ −2). Combining and removing the repeated value, the distinct roots are 4, −2, and 2. Product = 4 × (−2) × 2 = −16.
Answer: Product = −16
Q7. [CAT 2019, Slot 2] The equation x² + bx + c = 0 has two roots, 4a and 3a, where a is an integer. Find a possible value of b² + c.
Solution: Sum of roots: 4a + 3a = 7a = −b → b = −7a. Product of roots: (4a)(3a) = 12a² = c. So b² + c = 49a² + 12a² = 61a². This must equal one of the given answer choices for some integer a. Testing a = 3: 61(3)² = 61 × 9 = 549, and a = 3 is indeed an integer, so this works.
Answer: b² + c = 549 (when a = 3)
Q8. [CAT 2020, Slot 1] How many distinct positive integer-valued solutions exist for the equation (x² − 7x + 11)^(x² − 13x + 42) = 1?
Solution: An expression of the form (base)^(exponent) equals 1 in exactly three scenarios: (i) base = 1, for any exponent; (ii) exponent = 0, provided base ≠ 0; (iii) base = −1, provided the exponent is even.
Case (i): x² − 7x + 11 = 1 → x² − 7x + 10 = 0 → (x−2)(x−5) = 0 → x = 2, 5.
Case (ii): x² − 13x + 42 = 0 → (x−6)(x−7) = 0 → x = 6, 7 (checking the base is non-zero at both points — it is).
Case (iii): x² − 7x + 11 = −1 → x² − 7x + 12 = 0 → (x−3)(x−4) = 0 → x = 3, 4. Checking the exponent is even at both: at x = 3, exponent = 9−39+42 = 12 (even) ✓; at x = 4, exponent = 16−52+42 = 6 (even) ✓. Collecting all distinct values: {2, 3, 4, 5, 6, 7} — six values in total.
Answer: 6 solutions
Q9. [CAT 2020, Slot 1] Find the number of distinct real roots of the equation (x + 1/x)² − 3(x + 1/x) + 2 = 0.
Solution: Substitute y = x + 1/x: y² − 3y + 2 = 0 → (y−1)(y−2) = 0 → y = 1 or y = 2. For y = 1: x + 1/x = 1 → x² − x + 1 = 0 → discriminant = 1 − 4 = −3 < 0, so no real roots here. For y = 2: x + 1/x = 2 → x² − 2x + 1 = 0 → (x−1)² = 0 → x = 1 (a single repeated value). So across both cases, there is exactly one distinct real value of x.
Answer: 1 distinct real root
Q10. [CAT 2025, Slot 2] If 9^(x² + 2x − 3) − 4·3^(x² + 2x − 2) + 27 = 0, find the product of all possible values of x.
Solution: Let t = x² + 2x − 3. Then 9^t = 3^(2t), and the middle term 3^(x²+2x−2) = 3^(t+1) = 3·3^t. Substituting y = 3^t, the equation becomes y² − 4·3·y + 27 = 0 → y² − 12y + 27 = 0 → (y−3)(y−9) = 0 → y = 3 or y = 9, giving 3^t = 3 → t = 1, or 3^t = 9 → t = 2. For t = 1: x² + 2x − 3 = 1 → x² + 2x − 4 = 0. Discriminant = 4 + 16 = 20 > 0 (real roots); product of these two roots = −4/1 = −4. For t = 2: x² + 2x − 3 = 2 → x² + 2x − 5 = 0. Discriminant = 4 + 20 = 24 > 0 (real roots); product of these two roots = −5/1 = −5. Product of all four real values of x = (−4) × (−5) = 20.
Answer: Product = 20
Part 2: CAT-Pattern Quadratic Equations Practice Questions (Q11–Q50)
These 40 questions are original, written in CAT style and difficulty to build the same skills the PYQs above test. They are practice material, not attributed to any specific CAT year.
Section A: Basic Factorisation (Q11–Q17)
Q11. Solve for x: x² − 5x + 6 = 0
Solution: Split the middle term: x² − 2x − 3x + 6 = 0 → x(x−2) − 3(x−2) = 0 → (x−2)(x−3) = 0.
Answer: x = 2, 3
Q12. Solve for x: x² − 7x + 12 = 0
Solution: (x − 3)(x − 4) = 0.
Answer: x = 3, 4
Q13. Solve for x: 2x² − 3x − 2 = 0
Solution: Split: 2x² + x − 4x − 2 = 0 → x(2x+1) − 2(2x+1) = 0 → (2x+1)(x−2) = 0.
Answer: x = −1/2, 2
Q14. Solve for x: x² + x − 12 = 0
Solution: (x + 4)(x − 3) = 0.
Answer: x = −4, 3
Q15. Solve for x: 3x² − 5x + 2 = 0
Solution: 3x² − 3x − 2x + 2 = 0 → 3x(x−1) − 2(x−1) = 0 → (3x−2)(x−1) = 0.
Answer: x = 2/3, 1
Q16. Solve for x: x² − 2x − 8 = 0
Solution: (x − 4)(x + 2) = 0.
Answer: x = 4, −2
Q17. Solve for x: 6x² − x − 1 = 0
Solution: Discriminant = 1 + 24 = 25. x = (1 ± 5)/12.
Answer: x = 1/2, −1/3
Section B: Sum & Product of Roots (Q18–Q23)
Q18. Find the sum and product of the roots of x² − 7x + 10 = 0.
Solution: Using α + β = −b/a and αβ = c/a directly.
Answer: Sum = 7, Product = 10
Q19. If α, β are the roots of 2x² − 3x − 5 = 0, find α² + β².
Solution: α + β = 3/2, αβ = −5/2. α² + β² = (α+β)² − 2αβ = 9/4 + 5 = 29/4.
Answer: 29/4
Q20. If α, β are the roots of x² − 6x + 8 = 0, find 1/α + 1/β.
Solution: 1/α + 1/β = (α+β)/αβ = 6/8 = 3/4.
Answer: 3/4
Q21. If α, β are the roots of x² − 4x + 1 = 0, find α³ + β³.
Solution: α + β = 4, αβ = 1. α³ + β³ = (α+β)³ − 3αβ(α+β) = 64 − 12 = 52.
Answer: 52
Q22. If one root of x² − kx + 12 = 0 is 3, find k and the other root.
Solution: Substituting x = 3: 9 − 3k + 12 = 0 → k = 7. Since product of roots = 12, the other root = 12/3 = 4.
Answer: k = 7, other root = 4
Q23. If the roots of 3x² + (2k − 1)x + (k − 5) = 0 are reciprocals of each other, find k.
Solution: Roots are reciprocal ⟹ product of roots = 1 ⟹ (k − 5)/3 = 1 ⟹ k = 8.
Answer: k = 8
Section C: Nature of Roots — Discriminant (Q24–Q29)
Q24. Determine the nature of the roots of x² − 4x + 4 = 0.
Solution: D = 16 − 16 = 0.
Answer: Real and equal roots
Q25. Determine the nature of the roots of x² + 2x + 5 = 0.
Solution: D = 4 − 20 = −16 < 0.
Answer: No real roots (complex/imaginary roots)
Q26. Determine the nature of the roots of 2x² − 7x + 3 = 0.
Solution: D = 49 − 24 = 25, a perfect square.
Answer: Real, rational and distinct roots
Q27. Find the value(s) of k for which x² − kx + 9 = 0 has equal roots.
Solution: D = 0 ⟹ k² − 36 = 0 ⟹ k = ±6.
Answer: k = ±6
Q28. For what values of k does kx² + 2x + 1 = 0 have two real roots?
Solution: For a genuine quadratic, k ≠ 0. D ≥ 0 ⟹ 4 − 4k ≥ 0 ⟹ k ≤ 1.
Answer: k ≤ 1, k ≠ 0
Q29. For what value(s) of m does x² − (m + 3)x + (m + 6) = 0 have equal roots?
Solution: D = (m + 3)² − 4(m + 6) = 0 ⟹ m² + 2m − 15 = 0 ⟹ (m + 5)(m − 3) = 0.
Answer: m = −5 or m = 3
Section D: Forming Quadratic Equations (Q30–Q33)
Q30. Form the quadratic equation whose roots are the reciprocals of the roots of 2x² − 3x + 1 = 0.
Solution: Original sum = 3/2, product = 1/2. New sum = (α+β)/(αβ) = 3, new product = 1/(αβ) = 2.
Answer: x² − 3x + 2 = 0
Q31. Form the quadratic equation whose roots are the squares of the roots of x² − 5x + 6 = 0.
Solution: Roots of original equation: 2, 3. Squares: 4, 9. Sum = 13, product = 36.
Answer: x² − 13x + 36 = 0
Q32. If α, β are the roots of x² + 3x + 1 = 0, form the equation whose roots are α/β and β/α.
Solution: α + β = −3, αβ = 1. α/β + β/α = (α² + β²)/αβ = (9 − 2)/1 = 7. Product = 1.
Answer: x² − 7x + 1 = 0
Q33. Form the quadratic equation whose roots exceed the roots of x² − 3x + 2 = 0 by 2.
Solution: Roots of original equation: 1, 2. New roots: 3, 4. Sum = 7, product = 12.
Answer: x² − 7x + 12 = 0
Section E: CAT-Style Word Problems (Q34–Q37)
Q34. The product of two consecutive positive integers exceeds their sum by 89. Find the integers.
Solution: Let integers be n and n+1. n(n+1) − (2n+1) = 89 ⟹ n² − n − 90 = 0 ⟹ (n−10)(n+9) = 0 ⟹ n = 10.
Answer: 10 and 11
Q35. Two numbers differ by 3 and their product is 88. Find the numbers.
Solution: Let numbers be x and x+3. x(x+3) = 88 ⟹ x² + 3x − 88 = 0 ⟹ D = 361 = 19² ⟹ x = 8.
Answer: 8 and 11
Q36. The sum of a positive number and its reciprocal is 26/5. Find the number.
Solution: x + 1/x = 26/5 ⟹ 5x² − 26x + 5 = 0 ⟹ D = 576 = 24² ⟹ x = (26 ± 24)/10.
Answer: x = 5 or x = 1/5
Q37. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the original speed.
Solution: Let speed = x km/h. 360/x − 360/(x+5) = 1 ⟹ 1800 = x² + 5x ⟹ x² + 5x − 1800 = 0 ⟹ D = 7225 = 85² ⟹ x = 40.
Answer: 40 km/h
Section F: Higher-Degree Equations Reducible to Quadratic (Q38–Q41)
Q38. Solve for x: x⁴ − 13x² + 36 = 0
Solution: Substitute y = x²: y² − 13y + 36 = 0 ⟹ (y−4)(y−9) = 0 ⟹ y = 4, 9.
Answer: x = ±2, ±3
Q39. Solve for x: (x² − 3x)² − 2(x² − 3x) − 8 = 0
Solution: Substitute y = x² − 3x: y² − 2y − 8 = 0 ⟹ (y−4)(y+2) = 0 ⟹ y = 4 or y = −2. For y = 4: x² − 3x − 4 = 0 ⟹ x = 4, −1. For y = −2: x² − 3x + 2 = 0 ⟹ x = 1, 2.
Answer: x = −1, 1, 2, 4
Q40. Solve for x: 2²ˣ − 3·2ˣ + 2 = 0
Solution: Substitute y = 2ˣ: y² − 3y + 2 = 0 ⟹ (y−1)(y−2) = 0 ⟹ y = 1 or y = 2.
Answer: x = 0, 1
Q41. Solve for x: √(x + 7) = x − 5
Solution: Squaring: x + 7 = x² − 10x + 25 ⟹ x² − 11x + 18 = 0 ⟹ (x−9)(x−2) = 0 ⟹ x = 9 or 2. Checking x = 2: √9 = 3 ≠ (2−5) = −3, rejected. Checking x = 9: √16 = 4 = (9−5) ✓.
Answer: x = 9 (x = 2 is an extraneous root)
Section G: Quadratic Equations with Modulus & Inequalities (Q42–Q45)
Q42. Solve for x: |x² − 5x + 6| = 2
Solution: Case 1: x² − 5x + 6 = 2 ⟹ x² − 5x + 4 = 0 ⟹ x = 1, 4. Case 2: x² − 5x + 6 = −2 ⟹ x² − 5x + 8 = 0 ⟹ D = −7 < 0, no real solutions. Answer: x = 1, 4
Q43. Solve the inequality: x² − 5x + 6 < 0
Solution: Roots are 2 and 3. Since the coefficient of x² is positive, the expression is negative between the roots.
Answer: 2 < x < 3
Q44. Solve the inequality: x² − x − 6 ≥ 0
Solution: Roots are −2 and 3. Since the parabola opens upward, the expression is non-negative outside the roots.
Answer: x ≤ −2 or x ≥ 3
Q45. Solve for x: |x − 3|² − 4|x − 3| + 3 = 0
Solution: Substitute y = |x−3|: y² − 4y + 3 = 0 ⟹ (y−1)(y−3) = 0 ⟹ y = 1 or 3. y=1 ⟹ x = 4, 2. y=3 ⟹ x = 6, 0.
Answer: x = 0, 2, 4, 6
Section H: Advanced / TITA-Style Questions (Q46–Q50)
Q46. If the roots of ax² + bx + c = 0 are in the ratio 2 : 3, prove the relation connecting a, b and c.
Solution: Let roots be 2k and 3k. Sum = 5k = −b/a ⟹ k = −b/5a. Product = 6k² = c/a. Substituting k: 6b²/25a² = c/a.
Answer: 6b² = 25ac
Q47. If the equation x² − (a+1)x + (a−1) = 0 has roots that are equal in magnitude but opposite in sign, find a.
Solution: Roots equal in magnitude but opposite in sign ⟹ sum of roots = 0 ⟹ a + 1 = 0.
Answer: a = −1
Q48. Find the value(s) of k for which x² + 2(k+2)x + 9k = 0 has equal roots.
Solution: D = 0 ⟹ 4(k+2)² − 36k = 0 ⟹ (k+2)² − 9k = 0 ⟹ k² − 5k + 4 = 0 ⟹ (k−1)(k−4) = 0.
Answer: k = 1, 4
Q49. Find the number of real roots of (x² + x + 1)(x² + x + 2) = 12.
Solution: Let y = x² + x. (y+1)(y+2) = 12 ⟹ y² + 3y − 10 = 0 ⟹ (y+5)(y−2) = 0 ⟹ y = −5 or y = 2. For y = 2: x² + x − 2 = 0 ⟹ x = 1, −2 (2 real roots). For y = −5: x² + x + 5 = 0 ⟹ D = −19 < 0 (no real roots).
Answer: 2 real roots
Q50. Find the number of integer values of x satisfying x² − 5|x| + 6 = 0.
Solution: Substitute t = |x|, t ≥ 0: t² − 5t + 6 = 0 ⟹ (t−2)(t−3) = 0 ⟹ t = 2 or 3. So |x| = 2 gives x = ±2, and |x| = 3 gives x = ±3.
Answer: 4 integer solutions (x = −3, −2, 2, 3)
Common Mistakes Students Make in CAT Quadratic Equations
| Mistake | Why It Costs Marks | Fix |
|---|---|---|
| Forgetting the third case of aᵇ = 1 (a = −1, b even) | Directly causes wrong answers on questions like CAT 2020’s Q8 above | Always check all three cases: base = 1, exponent = 0, or base = −1 with even exponent |
| Not verifying roots after squaring radical equations | Leads to selecting an extraneous root as the final answer | Always substitute the final answer back into the original equation |
| Solving for actual roots when only α² + β² or α³ + β³ is asked | Slower and more error-prone | Use sum-and-product identities directly |
| Ignoring the case a = 0 in “for what value of k” questions | Missing valid cases where the equation becomes linear | Always state a ≠ 0 as a condition, or check the linear case separately |
| Mixing up “roots differ by n” with “roots in ratio n” | Leads to setting up the wrong two equations | Read word problems twice; identify whether it’s a difference or a ratio condition |
| Sign errors in the modulus cases | Missing one branch of the solution set | Always write both Case 1 (positive) and Case 2 (negative) explicitly |
How to Practice Quadratic Equations for CAT 2026
Reading solved questions builds understanding, but CAT rewards speed under exam-like pressure. Once you’re comfortable with all 50 questions above:
- Take a free CAT mock test on CATMOCK to see how quadratic equations questions appear alongside the rest of the Quant section, under real sectional timing.
- Browse CATMOCK’s CAT preparation blogs for topic-wise practice sets on Number Systems, Logarithms, Time-Speed-Distance and other Algebra areas that build on the same root-finding logic.
- Re-attempt the Quick Answer Key section above without looking at the solutions — this single step reveals which sub-topic (factorisation, nature of roots, or word problems) needs more practice before CAT 2026.
FAQs on CAT Quadratic Equations Questions
Q1. How many quadratic equations questions are asked in CAT?
CAT typically asks 0–2 direct questions on quadratic equations each year, though the concept supports several more questions across Number Systems, Inequalities, and Functions.
Q2. Are the CAT PYQs in this article real questions from CAT exams?
Yes. The 10 questions in Part 1 are verified previous year questions, each labelled with its confirmed CAT year and slot (1997, 2001, 2003, 2017, 2019, 2020 and 2025). They are written in original wording to avoid copying exact exam text, but every number, condition and final answer matches the original question.
Q3. What is the easiest way to solve CAT quadratic equations quickly?
Check if the discriminant (b² − 4ac) is a perfect square first — if it is, factorise by splitting the middle term instead of using the full quadratic formula, which saves valuable time.
Q4. What formulas should I memorise for CAT quadratic equations?
At minimum: the quadratic formula, discriminant conditions for nature of roots, sum of roots (−b/a), product of roots (c/a), and the identities for α² + β² and α³ + β³.
Q5. Can a quadratic equation have more than two roots?
No. A genuine quadratic equation (degree 2) always has exactly two roots — real or complex, distinct or equal — as per the Fundamental Theorem of Algebra.
Q6. How do I solve higher-degree equations like x⁴ − 13x² + 36 = 0 for CAT?
Use substitution. Replace x² with a new variable (say y) to reduce the equation to a standard quadratic in y, solve for y, and then back-substitute to find x.
Q7. Is there negative marking on quadratic equations questions in CAT?
Yes, if the question is in MCQ format: −1 mark for a wrong answer and +3 for a correct one. TITA (Type-In-The-Answer) questions carry no negative marking.
Conclusion
Quadratic equations may carry a modest direct weightage in CAT, but the concepts behind them — factorisation, sum-and-product shortcuts, discriminant analysis, and substitution — resurface constantly across the Quant section, as the verified PYQs from 1997 through 2025 in this article show. The fastest way to convert this practice into exam-day marks is to attempt it under timed conditions, review every mistake immediately, and repeat with a fresh mock. That repetition, more than any single formula, is what separates a 90th-percentile Quant score from a 99th-percentile one.
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