Quadratic Equation Solver – Find Roots with Steps

Enter coefficients a, b, c to compute roots with step-by-step explanation and discriminant analysis.

Derivation & Formula

A quadratic equation ax² + bx + c = 0 can be solved by completing the square or using the quadratic formula. Completing the square leads to the formula x = (-b ± √(b² - 4ac)) / (2a). The term under the square root, D = b² - 4ac, is called the discriminant and tells us whether the roots are real or complex.

Worked Examples

  1. For a=1, b=-3, c=2 ⇒ D = 1 ⇒ roots: x=1 and x=2 (distinct real roots).
  2. For a=1, b=2, c=1 ⇒ D = 0 ⇒ root: x=-1 (repeated root).
  3. For a=1, b=0, c=1 ⇒ D = -4 ⇒ roots: x = ±i (complex roots).

Understanding Quadratic Equations (A concise guide)

Quadratic equations are foundational in algebra and appear across physics, engineering and economics. Mastering how to solve them quickly and accurately is essential for students preparing for board exams and competitive tests such as JEE. The quadratic formula offers a universal method, but understanding discriminant behaviour and edge cases (linear reduction when a=0) helps avoid mistakes during timed tests.

When using the formula, always compute the discriminant first: D = b² − 4ac. If D > 0, there are two real and distinct roots; if D = 0, there is one repeated real root; and if D < 0, the roots are complex conjugates. In practice, small rounding errors can occur with floating-point arithmetic; for hand calculations prefer exact arithmetic (fractions or surds) when possible.

Practical tips: simplify coefficients (divide by common factors) before applying the formula, double-check signs carefully, and for integer-coefficient quadratics look for factorable patterns (e.g., when product ac and sum b yield integer roots). For graphing, the vertex formula x = −b/(2a) gives the axis of symmetry, and the value f(−b/(2a)) gives the minimum/maximum depending on the sign of a.

This tool provides step-by-step outputs to help you learn the method rather than just getting the answer. Use the example buttons to load typical cases, and compare manually computed roots with the tool to build confidence. For JEE-style problems, practice manipulating algebraic expressions to transform equations into solvable forms and identify when substitution or factoring is faster than the quadratic formula.

For more calculators and chapter-based toolkits, visit our Math hub or the JEE hub for focused practice sets. If you want a solver that outputs exact surd forms instead of decimal approximations, I can add symbolic output in the next iteration.

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