Quadratic Equations

Overview

Quadratic equations form a foundational topic in algebra that appears consistently in Assam TET Paper II. A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Understanding how to find roots and analyse their nature is essential for solving problems quickly in the exam.

This topic connects directly to other areas like coordinate geometry (parabolas), physics (projectile motion), and practical problems involving area and optimization. For TET, you must be comfortable with the standard form, factorization methods, the quadratic formula, and the discriminant. Questions typically test your ability to find roots, determine their nature, and apply relationships between roots and coefficients.

Mastery of quadratic equations demonstrates algebraic reasoning—a key competency for upper primary mathematics teachers who must explain these concepts clearly to students.

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 crucial—without it, 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 or complex).

• **Discriminant (D)**: The expression D = b² − 4ac determines the nature of roots without actually solving the equation. This is a powerful shortcut for MCQs.

• **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 (complex/imaginary roots)

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

• **Methods to solve**: Factorization, completing the square, and quadratic formula. For exams, factorization is fastest when applicable; formula is the universal backup.

• **Forming equations from roots**: If roots are given as α and β, the equation is x² − (α + β)x + αβ = 0.

Formulas / Key Facts

| Formula | Context | |---------|---------| | ax² + bx + c = 0 | Standard form (a ≠ 0) | | x = (−b ± √(b² − 4ac)) / 2a | Quadratic formula—universal method | | D = b² − 4ac | Discriminant—determines nature of 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 |

**Quick facts to remember**:

• When D is a perfect square and coefficients are rational, roots are rational.
• Equal roots mean the quadratic is a perfect square trinomial.
• If c = 0, one root is always zero.
• If b = 0, roots are equal in magnitude but opposite in sign.

Worked Examples

**Example 1**: Find the roots of x² − 5x + 6 = 0.

*Solution by factorization*:

• Find two numbers whose product is 6 and sum is −5.
• Numbers are −2 and −3.
• x² − 2x − 3x + 6 = 0
• x(x − 2) − 3(x − 2) = 0
• (x − 2)(x − 3) = 0
• Roots: x = 2 and x = 3

---

**Example 2**: Determine the nature of roots of 2x² + 4x + 5 = 0.

*Solution*:

• Here a = 2, b = 4, c = 5
• D = b² − 4ac = 16 − 40 = −24
• Since D < 0, the equation has no real roots.

---

**Example 3**: If one root of x² − 6x + k = 0 is 4, find k and the other root.

*Solution*:

• Substitute x = 4: 16 − 24 + k = 0 → k = 8
• Equation becomes x² − 6x + 8 = 0
• Sum of roots = 6, so other root = 6 − 4 = 2
• Verification: Product = 4 × 2 = 8 = k ✓

---

**Example 4**: Form a quadratic equation whose roots are 3 and −2.

*Solution*:

• Sum of roots = 3 + (−2) = 1
• Product of roots = 3 × (−2) = −6
• Equation: x² − (sum)x + (product) = 0
• Answer: x² − x − 6 = 0

Common Mistakes

• **Forgetting that a ≠ 0** → If a = 0 in ax² + bx + c = 0, it becomes a linear equation bx + c = 0, not quadratic. Always verify.

• **Sign error in sum of roots** → Students write α + β = b/a instead of −b/a. Remember the negative sign—it's the most common slip in exams.

• **Calculating discriminant incorrectly** → Writing D = b² + 4ac instead of b² − 4ac. The subtraction is critical.

• **Assuming D > 0 means rational roots** → D > 0 only guarantees real roots. For rational roots, D must also be a perfect square (when coefficients are rational).

• **Ignoring the ± in quadratic formula** → The formula gives two roots. Writing only one root loses half the answer.

Quick Reference

• Standard form: ax² + bx + c = 0 (a ≠ 0)
• Quadratic formula: x = (−b ± √D) / 2a where D = b² − 4ac
• D > 0 → real distinct; D = 0 → real equal; D < 0 → no real roots
• Sum of roots = −b/a; Product of roots = c/a
• To form equation from roots α, β: x² − (α + β)x + αβ = 0
• Always check your answer by substituting roots back into the original equation