Quadratic Equation

Introduction to the Quadratic Equation Calculator

The Quadratic Equation Calculator solves ax^2 + bx + c = 0 using the quadratic formula x = (-b plus or minus sqrt(b^2 - 4ac))/(2a). It returns both roots, the discriminant D = b^2 - 4ac, and indicates whether the roots are real and distinct (D > 0), real and equal (D = 0), or complex conjugate (D < 0).

Indian students from CBSE Class 10 chapter 4 (Quadratic Equations) to JEE Main and JEE Advanced rely on this tool. It is also useful for board exam-pattern factorisation problems, completing the square method, and word problems involving age, speed, area, and revenue maximisation.

You enter coefficients a, b and c. The calculator returns both roots in exact and decimal form, the discriminant, vertex (-b/2a, -D/4a) for parabola sketching, sum of roots (-b/a) and product of roots (c/a). Vieta's formulas are also shown for verification.

Who Should Use This Quadratic Equation Calculator

Class 10 CBSE and ICSE students in Delhi practicing NCERT chapter 4 and RD Sharma exercises use it to verify factorisation, completing the square, and quadratic formula answers before pre-boards.

JEE Main candidates in Kota solving 4-mark single correct quadratic problems on common roots, equal roots conditions, and parametric quadratics need fast verification under exam time pressure.

State board students in Tamil Nadu and Maharashtra preparing for SSC use it on Marathi and Tamil medium textbook problems involving Pythagorean theorem and quadratic age problems.

Engineering aspirants in Lucknow studying conic sections and parabola questions for JEE Advanced need vertex and discriminant analysis for optimisation problems.

College students in Bengaluru taking first-year engineering math (M1, M2 papers) apply quadratic theory to eigenvalue problems, second-order ODEs, and signal processing fundamentals.

Tips for Solving Quadratics

Smart Quadratic Equation Tips

Always try factorisation first if a, b, c are small integers. Look for two numbers that multiply to ac and add to b. For x^2 - 5x + 6 = 0, the pair is (-2, -3), giving (x-2)(x-3) = 0.

Use the discriminant to decide method before solving. If D = b^2 - 4ac is a perfect square, factorisation works. If D > 0 but not a perfect square, use the quadratic formula. If D < 0, switch to complex roots.

For CBSE Class 10 board exams, completing the square is required for 3-mark questions even when factorisation is faster. Practice the algorithm: divide by a, move c to RHS, add (b/2a)^2 to both sides.

Vieta's formulas save time: sum of roots = -b/a and product = c/a. If asked for roots and you find sum = 5, product = 6, the roots are 2 and 3 directly, skipping the formula entirely.

For JEE problems involving common roots of two quadratics, equate ratios: a1/a2 = b1/b2 = c1/c2 for both roots common, or use the resultant formula for exactly one common root.

Formula Explanation

Core Quadratic Formula

x = (-b plus or minus sqrt(b^2 - 4ac)) / (2a)

D = b^2 - 4ac (Discriminant)

Sum of roots = -b/a, Product of roots = c/a (Vieta's)

Where:

• a, b, c = coefficients of ax^2 + bx + c = 0
• • a not equal to 0 (otherwise it is linear)
• • D > 0: two distinct real roots
• • D = 0: one repeated real root (-b/2a)
• • D < 0: two complex conjugate roots

Example: For 2x^2 - 7x + 3 = 0, D = 49 - 24 = 25, roots = (7 plus or minus 5)/4 = 3 or 0.5. Sum = 3.5 = 7/2 = -b/a, Product = 1.5 = 3/2 = c/a. Verified.

Quadratic Quick Reference Table

Equation	a, b, c	Discriminant	Roots
x^2 - 5x + 6 = 0	1, -5, 6	1	2, 3
x^2 - 4x + 4 = 0	1, -4, 4	0	2, 2
x^2 + 4x + 5 = 0	1, 4, 5	-4	-2 plus or minus i
2x^2 - 7x + 3 = 0	2, -7, 3	25	3, 0.5
3x^2 + 5x - 2 = 0	3, 5, -2	49	1/3, -2

Real-World Example

Example: Arjun's Pre-Board Quadratic Practice

Meet Arjun, 15, a Class 10 CBSE student from Chennai preparing for board exams in March 2027. His teacher gives him the equation 2x^2 - 7x + 3 = 0 with instructions to solve by factorisation, formula and verify with Vieta's.

Step 1: Arjun identifies a = 2, b = -7, c = 3. He computes ac = 6 and looks for factors of 6 that add to -7: the pair is -1 and -6.

Step 2: He splits the middle term: 2x^2 - 6x - x + 3 = 0, then factors as 2x(x - 3) - 1(x - 3) = (2x - 1)(x - 3) = 0. Roots: x = 1/2 or x = 3.

Step 3: He verifies with the formula. D = 49 - 24 = 25. x = (7 plus or minus 5)/4 = 3 or 1/2. Sum check: 3 + 0.5 = 3.5 = 7/2 = -b/a. Product check: 1.5 = 3/2 = c/a.

Result: Arjun's roots are 1/2 and 3, verified by three methods. He writes the full working for full marks in his board pattern question, scoring 4/4 on this problem.

Frequently Asked Questions About Quadratic Equations

Indian students often ask when to use factorisation versus completing the square versus the quadratic formula, how to handle parametric quadratics in JEE problems, what the discriminant tells about the parabola's shape, and how to verify roots using Vieta's formulas. The FAQ below covers each method with CBSE board-pattern examples, JEE problem-set tips, and RD Sharma reference exercises.