Quadratic Equations

Overview

Quadratic equations form a cornerstone topic in upper-primary and secondary mathematics, appearing consistently in JTET Paper II. A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable is 2. This topic tests both computational skills (finding roots) and conceptual understanding (nature of roots, forming equations).

For JTET, you must master three core competencies: solving quadratic equations using different methods, determining the nature of roots without actually solving, and applying these concepts to word problems. Questions typically involve factorisation, the quadratic formula, and interpreting the discriminant. Real-world applications—such as area problems, age problems, and number puzzles—are common exam themes.

Understanding quadratic equations also builds the foundation for teaching algebraic thinking to upper-primary students, making this topic pedagogically significant beyond mere problem-solving.

Key Concepts

• **Standard Form**: A quadratic equation is written as ax² + bx + c = 0, where a ≠ 0. The coefficients a, b, and c are real numbers, and x is the variable.

• **Roots/Solutions**: The values of x that satisfy the equation are called roots. 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 solving the equation. This is a frequently tested concept.

• **Sum and Product of Roots**: If α and β are roots, then α + β = −b/a and αβ = c/a. These relationships help form equations when roots are given.

• **Methods of Solving**: Three primary methods exist—factorisation (splitting middle term), completing the square, and the quadratic formula. Factorisation is fastest when applicable.

• **Graphical Interpretation**: The graph of y = ax² + bx + c is a parabola. Roots are the x-intercepts where the parabola crosses the x-axis.

Formulas / Key Facts

| Formula | Context | |---------|---------| | ax² + bx + c = 0 | Standard form of quadratic equation | | x = (−b ± √(b² − 4ac)) / 2a | Quadratic formula to find both roots | | D = b² − 4ac | Discriminant formula | | D > 0 | Two distinct real roots | | D = 0 | Two equal real roots (one repeated root) | | D < 0 | No real roots (imaginary roots) | | α + β = −b/a | Sum of roots | | αβ = c/a | Product of roots | | x² − (sum)x + (product) = 0 | Forming equation when roots are known |

**Key Fact**: For JTET level, discriminant questions and factorisation-based problems dominate. Complex/imaginary roots are typically only mentioned conceptually, not computed.

Worked Examples

### Example 1: Solving by Factorisation **Solve: x² − 5x + 6 = 0**

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

Step 2: Split the middle term. x² − 2x − 3x + 6 = 0

Step 3: Factor by grouping. x(x − 2) − 3(x − 2) = 0 (x − 2)(x − 3) = 0

Step 4: Set each factor to zero. x = 2 or x = 3

**Answer**: Roots are 2 and 3.

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### Example 2: 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 the 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**: Roots are −1/2 and −3.

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### Example 3: Nature of Roots (Without Solving) **Determine the nature of roots: 3x² − 2x + 5 = 0**

a = 3, b = −2, c = 5

D = b² − 4ac = 4 − 60 = −56

Since D < 0, the equation has **no real roots**.

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### Example 4: Application Problem **The product of two consecutive positive integers is 132. Find them.**

Let the integers be x and x + 1. x(x + 1) = 132 x² + x − 132 = 0

Find factors of 132 with difference 1: 12 and 11 Split: x² + 12x − 11x − 132 = 0 x(x + 12) − 11(x + 12) = 0 (x + 12)(x − 11) = 0

x = 11 or x = −12 (rejected as integers are positive)

**Answer**: The integers are 11 and 12.

Common Mistakes

• **Forgetting a ≠ 0**: Students sometimes accept equations where the coefficient of x² becomes zero after simplification. Always verify a ≠ 0; otherwise, it's a linear equation.

• **Sign errors in discriminant**: When b is negative, students forget that (−b)² is positive. Example: For b = −4, b² = 16, not −16.

• **Misapplying sum/product formulas**: Students often write sum of roots = b/a instead of −b/a. Remember the negative sign in the sum formula.

• **Incomplete factorisation**: After splitting the middle term, students sometimes stop at grouping without extracting the common binomial factor.

• **Ignoring context in word problems**: Negative roots may be mathematically valid but contextually meaningless (e.g., negative age, negative length). Always verify answers against the problem's real-world constraints.

• **Calculation errors in the quadratic formula**: The denominator is 2a, not 2. For 3x² + 5x + 1 = 0, divide by 6, not 2.

Quick Reference

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

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

• **D > 0 → 2 distinct roots; D = 0 → 2 equal roots; D < 0 → no real roots**

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

• **Factorisation**: Find two numbers with product = ac and sum = b

• **To form equation from roots α, β**: x² − (α + β)x + αβ = 0