Quadratic Equations

Overview

Quadratic equations form a cornerstone topic in upper-primary and secondary mathematics, appearing consistently in Bihar TET Paper II. A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. This topic bridges basic algebra with more advanced mathematical reasoning and has direct applications in real-world problems involving area, projectile motion, and optimization.

For Bihar TET, you must master three core skills: identifying quadratic equations, finding their roots using multiple methods, and applying these concepts to word problems. The syllabus specifically emphasizes "roots of quadratic equations and applications," so expect questions testing both computational ability and conceptual understanding of the nature of roots.

This topic connects closely with algebraic expressions and linear equations from your syllabus. Strong command here also supports geometry problems (area calculations) and science applications (motion under gravity).

Key Concepts

• **Standard Form**: A quadratic equation in variable x is written as ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. The condition a ≠ 0 is essential—if a = 0, the equation becomes linear.

• **Roots/Solutions**: The values of x that satisfy the equation are called roots or zeros. A quadratic equation has exactly two roots (which may be equal, distinct, or complex).

• **Discriminant (D)**: The expression D = b² − 4ac determines the nature of roots. This is the most important diagnostic tool for quadratic equations.

• **Nature of Roots Based on D**:
• D > 0: Two distinct real roots
• D = 0: Two equal real roots (one repeated root)
• D < 0: No real roots (roots are complex/imaginary)

• **Sum and Product of Roots**: If α and β are roots of ax² + bx + c = 0, then:
• Sum of roots (α + β) = −b/a
• Product of roots (αβ) = c/a

• **Methods to Solve**: Factorization, completing the square, and quadratic formula are the three standard methods. For TET, factorization and the quadratic formula are most frequently tested.

• **Forming Equations**: Given roots α and β, the quadratic equation is x² − (α + β)x + αβ = 0.

Formulas / Key Facts

| Formula | Context | |---------|---------| | ax² + bx + c = 0 | Standard form of quadratic equation | | x = (−b ± √(b² − 4ac)) / 2a | Quadratic formula (Shridharacharya's rule) | | D = b² − 4ac | Discriminant formula | | α + β = −b/a | Sum of roots | | αβ = c/a | Product of roots | | x² − (sum)x + (product) = 0 | Forming equation from roots |