Theory of Quadratic Equations

Welcome to your ultimate and comprehensive study guide for Theory of Quadratic Equations. Building directly upon the basics of quadratic equations, this chapter delves deeper into the nature of roots, the properties of cube roots of unity, the intricate relationships between roots and coefficients, and the formation of equations from given roots.

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1. The Discriminant and Nature of Roots

For any standard quadratic equation in the form ax² + bx + c = 0 (where a ≠ 0), the roots are given by the quadratic formula:
x = [-b ± √(b² – 4ac)] / 2a

The expression under the square root, b² – 4ac, is called the discriminant of the quadratic equation. The discriminant allows us to determine the exact nature of the roots without actually solving the equation:

• Real (Rational) and Equal: If b² – 4ac = 0.
• Real (Rational) and Unequal: If b² – 4ac > 0 and is a perfect square.
• Real (Irrational) and Unequal: If b² – 4ac > 0 and is not a perfect square.
• Imaginary (Complex Conjugates): If b² – 4ac < 0.

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2. Cube Roots of Unity and Their Properties

Let x be the cube root of unity, meaning x³ = 1. Solving x³ – 1 = 0 yields three distinct cube roots of unity:

• One real root: 1
• Two complex (imaginary) roots: ω = (-1 + √-3) / 2 and ω² = (-1 – √-3) / 2.

Fundamental Properties of Cube Roots of Unity:

• Each complex cube root of unity is the square of the other (ω² is the square of ω).
• The product of all three cube roots of unity is equal to one: 1 × ω × ω² = ω³ = 1.
• The sum of all three cube roots of unity is zero: 1 + ω + ω² = 0. Consequently, 1 + ω = -ω², 1 + ω² = -ω, and ω + ω² = -1.

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3. Relation Between Roots and Coefficients

If α and β are the two roots of the quadratic equation ax² + bx + c = 0, we can establish direct relationships between these roots and the coefficients (a, b, c) without solving the equation:

• Sum of the Roots (S): α + β = -b / a = -(Coefficient of x) / (Coefficient of x²).
• Product of the Roots (P): αβ = c / a = (Constant Term) / (Coefficient of x²).

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4. Formation of a Quadratic Equation

If the roots (α and β) of a quadratic equation are known, we can construct or form the original quadratic equation using the formula:
x² – (Sum of the Roots)x + (Product of the Roots) = 0
or simply: x² – Sx + P = 0.

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5. Step-by-Step Example: Determining Nature of Roots

Example: Using the discriminant, find the nature of the roots of 7x² + 8x + 1 = 0 and verify by solving.
Step-by-Step Solution:

• Step 1: Identify coefficients: Here, a = 7, b = 8, and c = 1.
• Step 2: Calculate the discriminant:
  Discriminant = b² – 4ac = (8)² – 4(7)(1) = 64 – 28 = 36.
• Step 3: Analyze the result: Since 36 is positive and a perfect square (6²), the roots are rational (real) and unequal.
• Step 4: Verify by factoring:
  7x² + 7x + x + 1 = 0 → 7x(x + 1) + 1(x + 1) = 0 → (x + 1)(7x + 1) = 0.
  Roots are x = -1 and x = -1/7, confirming they are real, rational, and unequal.

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Essential Conceptual Review Questions

Q1: What does a negative discriminant tell us about the graph or roots of a quadratic equation?
Answer: If the discriminant (b² – 4ac) is less than zero (negative), it means the quadratic equation has no real roots. Instead, its roots are imaginary (complex conjugates), and its parabolic graph will not intersect the x-axis.

Q2: Why is ω³ always equal to 1 in the theory of cube roots of unity?
Answer: By definition, ω is a cube root of unity, meaning when it is cubed (ω³), it satisfies the equation x³ = 1, making its value identically equal to 1.