Quadratic Equations

Overview

Quadratic equations form a cornerstone topic in the CG TET Paper II Mathematics section. These equations appear in various forms—direct solving, word problems, and questions testing the relationship between roots and coefficients. Mastering this topic builds a foundation for higher algebra and helps in solving real-world problems involving areas, projectile motion, and optimization.

For CG TET, you must be able to identify quadratic equations, solve them using multiple methods (factorisation, formula, completing the square), determine the nature of roots using the discriminant, and apply the sum-product relationships of roots. Questions typically test computational accuracy and conceptual clarity, so both speed and understanding matter equally.

Key Concepts

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

• **Roots/Solutions**: Values of x that satisfy the equation. 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 without actually solving the equation.

• **Methods of Solving**: Four standard approaches exist—factorisation, quadratic formula, completing the square, and graphical method. For exams, factorisation and formula are most frequently tested.

• **Sum and Product of Roots**: If α and β are roots, then α + β = −b/a and αβ = c/a. These relationships allow you to form equations or find expressions involving roots.

• **Symmetric Functions**: Expressions like α² + β², α³ + β³, and 1/α + 1/β can be computed using sum and product without finding individual roots.

Formulas / Key Facts

| Formula | Context | |---------|---------| | ax² + bx + c = 0 | Standard form of quadratic equation | | x = (−b ± √(b² − 4ac)) / 2a | Quadratic formula (Shreedharacharya's rule) | | D = b² − 4ac | Discriminant formula | | D > 0 → Two distinct real roots | Nature of roots | | D = 0 → Two equal real roots | Both roots are −b/2a | | D < 0 → No real roots | Roots are complex conjugates | | α + β = −b/a | Sum of roots | | αβ = c/a | Product of roots | | x² − (α + β)x + αβ = 0 | Equation with roots α and β | | α² + β² = (α + β)² − 2αβ | Derived symmetric function | | α − β = ±√D / a | Difference of roots |

Worked Examples

**Example 1: Solve by Factorisation**

Solve: x² − 5x + 6 = 0

Step 1: Find two numbers whose product is 6 and sum is −5. Numbers are −2 and −3 (since −2 × −3 = 6 and −2 + −3 = −5)

Step 2: Factorise. x² − 2x − 3x + 6 = 0 x(x − 2) − 3(x − 2) = 0 (x − 2)(x − 3) = 0

Step 3: Set each factor to zero. x = 2 or x = 3

**Answer**: x = 2, 3

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**Example 2: Solve Using Quadratic Formula**

Solve: 2x² + 7x + 3 = 0

Here a = 2, b = 7, c = 3

Step 1: Calculate discriminant. D = b² − 4ac = 49 − 24 = 25

Step 2: Apply formula. x = (−7 ± √25) / (2 × 2) x = (−7 ± 5) / 4

Step 3: Find both roots. x = (−7 + 5)/4 = −2/4 = −1/2 x = (−7 − 5)/4 = −12/4 = −3

**Answer**: x = −1/2, −3

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**Example 3: Nature of Roots**

For what value of k does x² + kx + 9 = 0 have equal roots?

For equal roots, D = 0 b² − 4ac = 0 k² − 4(1)(9) = 0 k² = 36 k = ±6

**Answer**: k = 6 or k = −6

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**Example 4: Sum and Product Application**

If α and β are roots of x² − 7x + 12 = 0, find α² + β².

Step 1: Find sum and product. α + β = 7 (using −b/a) αβ = 12 (using c/a)

Step 2: Use identity. α² + β² = (α + β)² − 2αβ α² + β² = 49 − 24 = 25

**Answer**: 25

Common Mistakes

• **Forgetting a ≠ 0**: Students sometimes accept equations like 0x² + 5x + 3 = 0 as quadratic. This is linear (5x + 3 = 0). Always verify the coefficient of x² is non-zero.

• **Sign errors in quadratic formula**: The formula has −b, not b. When b is negative, −b becomes positive. Write out the formula completely before substituting values.

• **Misinterpreting D = 0**: Students think D = 0 means "no roots." Actually, D = 0 means two equal (repeated) roots, not absence of roots. D < 0 means no real roots.

• **Wrong sum formula sign**: Sum of roots is −b/a, not b/a. The negative sign is frequently forgotten, leading to incorrect answers in root-based problems.

• **Incomplete factorisation**: When factorising, students sometimes stop at x(x − 5) = 6 and write x = 6 or x − 5 = 6. You must first bring everything to one side (equation = 0) before factorising.

• **Confusing roots with coefficients**: The roots α and β are solutions of the equation, not the values a, b, c. These are different entities serving different purposes.

Quick Reference

• Standard form: ax² + bx + c = 0 with a ≠ 0

• Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a

• D > 0: two distinct real roots; D = 0: equal roots; D < 0: no real roots

• Sum of roots = −b/a; Product of roots = c/a

• To form equation from roots: x² − (sum)x + (product) = 0

• Always factorise when the equation equals zero, not any other number