Quadratic Equations Practice Questions for CAT, CMAT & CUET 2026 — Basics with Solutions
- Jul 28, 2025
- 5 min read
Updated: Apr 13
Introduction
Quadratic equations are a fundamental concept in algebra that play a crucial role in various mathematical applications. Understanding how to solve these equations is essential for students, especially when preparing for exams. This blog aims to provide practice questions on quadratic equations, helping learners to strengthen their problem-solving skills and build confidence in their mathematical abilities. By mastering quadratic equations, students not only enhance their performance in exams but also gain valuable tools for tackling real-world problems in fields such as physics, engineering, and economics. Join us as we explore the significance of quadratic equations and offer a range of practice questions to aid your preparation.
Solve these Quadratic Equations by Formula:
2x2 + 4x + 2 = 0
2x2 + 5x + 2 = 0
2x2 + 7x + 5 = 0
2x2 + 11x + 5 = 0
2x2 + 12x + 10 = 0
2x2 + 7x + 6 = 0
2x2 - 7x + 3 = 0
2x2 + 9x + 10 = 0
2x2 - 9x + 4 = 0
2x2 - 5x + 3 = 0
x2 - 6x + 5 = 0
x2 + 8x + 12 = 0
x2 + 8x + 12 = 0
x2 + 13x - 14 = 0
x2 + 5x - 14 = 0
x2 + x - 20 = 0
2x2 + 11x + 5 = 0
2x2 + 7x + 5 = 0
2x2 + 9x + 10 = 0
2x2 - 9x + 4 = 0
2x2 - 6x + 4 = 0
2x2 - 10x + 8 = 0
2x2 - 8x + 8 = 0
2x2 - 15x + 7 = 0
2x2 - 9x + 7 = 0
x2 + 7x + 12 = 0
x2 + 10x + 16 = 0
x2 + 8x + 15 = 0
x2 + 12x + 20 = 0
x2 + 9x + 18 = 0
x2 + 12x + 32 = 0
2x2 - 10x + 8 = 0
2x2 + 6x + 4 = 0
2x2 - 15x + 7 = 0
2x2 - 8x + 8 = 0
2x2 - 9x + 7 = 0
2x2 - x - 21 = 0
2x2 - 12x - 14 = 0
Form a quadratic equation whose roots are 5 & -11
Form a quadratic equation whose roots are -11 & -13
Frequently Asked Questions — Quadratic Equations for CAT, CMAT & CUET 2026
What is a quadratic equation and how is it tested in CAT?
A quadratic equation is of the form ax²+bx+c=0 where a≠0. In CAT, quadratic equations appear in algebra topics — finding roots, nature of roots, max/min values, and quadratic inequalities. They also form the basis for higher-degree polynomial questions.
What is the discriminant and what does it tell us?
The discriminant is D = b²-4ac. If D>0, the equation has two distinct real roots. If D=0, it has two equal real roots. If D<0, it has no real roots (complex roots). CAT frequently tests the nature of roots using the discriminant.
What are Vieta's formulas for a quadratic equation?
For ax²+bx+c=0: Sum of roots (α+β) = -b/a, and Product of roots (αβ) = c/a. These formulas are tested directly in CAT, CMAT, and CUET questions involving sum and product of roots without actually solving the equation.
How do I find the maximum or minimum value of a quadratic expression?
For f(x)=ax²+bx+c: if a>0, the minimum value is (4ac-b²)/4a at x=-b/2a. If a<0, the maximum value is (4ac-b²)/4a. This concept frequently appears in CAT as a direct or applied question.
What are the most common mistakes students make in quadratic questions?
Common mistakes include: forgetting to check for D≥0 when roots must be real, confusing sum and product of roots formulas, not considering the sign of 'a' when finding max/min, and making sign errors when applying the quadratic formula.
How are quadratic equations tested in CMAT and CUET compared to CAT?
CMAT and CUET test more straightforward quadratic problems (find roots, use Vieta's, basic inequalities), while CAT questions are more conceptual and often combine quadratics with other algebra topics. The same preparation covers all three exams.
What is the quadratic formula and when should I use it?
The quadratic formula is x = (-b ± √(b²-4ac)) / 2a. Use it when the equation cannot be easily factored. However, in CAT exam conditions, try factoring first as it is faster. Practice factoring quadratics with integer roots.
How much time should I spend on quadratic equations while preparing for CAT?
Spend 3–4 days on quadratic equations — 1 day on theory (roots, discriminant, Vieta's, max/min), 2 days on 25–30 practice problems, and 1 day on mock questions. Quadratics also support preparation for inequalities and polynomials.
Frequently Asked Questions — Quadratic Equations for CAT & IPMAT 2026
What is a quadratic equation?
A quadratic equation is of the form ax² + bx + c = 0 where a ≠ 0. It has exactly two roots (real or complex). CAT tests quadratic equations in both direct form and as disguised higher-degree equations reducible to quadratics.
What is the quadratic formula?
x = [-b ± √(b² - 4ac)] / 2a. The term b² - 4ac is the discriminant (D). D > 0: two distinct real roots. D = 0: two equal real roots. D < 0: no real roots (complex roots). Memorising this is essential for CAT.
What are Vieta's formulas?
For ax² + bx + c = 0 with roots α and β: Sum of roots α + β = -b/a. Product of roots αβ = c/a. These formulas let you find expressions like α² + β² = (α+β)² - 2αβ without solving the equation directly — very useful in CAT.
How do I factorise a quadratic quickly?
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Example: x² + 5x + 6 = 0 → factors (x+2)(x+3) = 0 → roots x = -2, -3. For ax² + bx + c, use the splitting-the-middle-term method or apply the quadratic formula directly.
What is the nature of roots?
Discriminant D = b² - 4ac determines root nature. D > 0 and a perfect square: rational roots. D > 0 but not perfect square: irrational roots. D = 0: equal roots = -b/2a. D < 0: no real roots. CAT tests all these cases in MCQs.
How are quadratic equations used in CAT word problems?
Many CAT word problems reduce to quadratics: speed-distance, age problems, area problems, and number problems. Set up the equation from the given conditions, solve using factorisation or the formula, and check which root is valid in context.
What is the maximum/minimum value of a quadratic?
For f(x) = ax² + bx + c: if a > 0, minimum value = c - b²/4a at x = -b/2a. If a < 0, maximum value = c - b²/4a at x = -b/2a. CAT tests this in optimisation questions and inequality-based problems.
How do I solve quadratic inequalities?
For ax² + bx + c > 0: first find roots r₁ < r₂. If a > 0, solution is x < r₁ or x > r₂. For ax² + bx + c < 0 with a > 0: r₁ < x < r₂. Always check the sign of 'a' first before writing the solution set.
What are disguised quadratics in CAT?
Equations like x⁴ - 5x² + 6 = 0 or 2^(2x) - 5(2^x) + 6 = 0 are quadratics in disguise. Substitute t = x² or t = 2^x to get a standard quadratic in t, solve for t, then back-substitute to find x. CAT tests this in 1-2 questions per paper.
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