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Table of Contents

1. Quadratic Equation and Inequalities
   1.1. Roots of Quadratic Equations
   1.2. Formation of an Equation
   1.3. Common Concepts
   1.4. Special Cases
   1.5. Inequalities
2. Cubic Equations
3. Formation of an Equation
4. Common Concepts
5. Special Cases

1. Quadratic Equation and Inequalities

1.1. Roots of Quadratic Equations

• Definition: A quadratic equation is of the form $ ax^2 + bx + c = 0 $, where $ a \neq 0 $.
• Roots: The solutions to the equation are called roots.
• Sum and Product of Roots:
  • Sum: $ \alpha + \beta = - \dfrac{b}{a} $
  • Product: $ \alpha \beta = \dfrac{c}{a} $
• Examples:
  • If $ \alpha = 2 $ and $ \beta = 3 $, the equation is $ x^2 - 5x + 6 = 0 $.
  • If $ \alpha = -1 $ and $ \beta = 4 $, the equation is $ x^2 - 3x - 4 = 0 $.

1.2. Formation of an Equation

• General Form: If roots are $ \alpha $ and $ \beta $, the equation is $ x^2 - (\alpha + \beta)x + \alpha\beta = 0 $.
• Example:
  • Roots $ \alpha = 2 $, $ \beta = 3 $:
    $$ x^2 - (2 + 3)x + (2 \times 3) = 0 \implies x^2 - 5x + 6 = 0 $$

1.3. Common Concepts

• Discriminant: $ D = b^2 - 4ac $.
  • If $ D > 0 $: Two distinct real roots.
  • If $ D = 0 $: One real root (repeated).
  • If $ D < 0 $: Two complex conjugate roots.
• Nature of Roots: Determined by the discriminant.

1.4. Special Cases

• Perfect Square Trinomial:
  • Example: $ x^2 + 6x + 9 = 0 $, roots $ x = -3 $ (repeated).
• Equation with Zero Coefficients:
  • If $ a = 0 $, the equation becomes linear: $ bx + c = 0 $.

1.5. Inequalities

• Quadratic Inequalities: Solve $ ax^2 + bx + c > 0 $, $ ax^2 + bx + c < 0 $, etc.
• Steps:
  1. Find roots of the equation $ ax^2 + bx + c = 0 $.
  2. Determine the intervals based on the roots.
  3. Test each interval to determine where the inequality holds.
• Example:
  • Solve $ x^2 - 5x + 6 > 0 $.
  • Roots: $ x = 2 $, $ x = 3 $.
  • Test intervals: $ (-\infty, 2) $, $ (2, 3) $, $ (3, \infty) $.
  • Solution: $ x < 2 $ or $ x > 3 $.

2. Cubic Equations

• General Form: $ ax^3 + bx^2 + cx + d = 0 $, where $ a \neq 0 $.
• Roots: A cubic equation has three roots (real or complex).
• Rational Root Theorem: Possible rational roots are $ \pm \dfrac{p}{q} $, where $ p $ divides $ d $, and $ q $ divides $ a $.
• Example:
  • Equation: $ x^3 - 6x^2 + 11x - 6 = 0 $.
  • Possible roots: $ \pm1, \pm2, \pm3, \pm6 $.
  • Actual roots: $ x = 1, 2, 3 $.

3. Formation of an Equation

• From Roots: If roots are $ \alpha $ and $ \beta $, the equation is $ x^2 - (\alpha + \beta)x + \alpha\beta = 0 $.
• From Coefficients: If sum of roots $ S = \alpha + \beta $ and product $ P = \alpha\beta $, then the equation is $ x^2 - Sx + P = 0 $.
• Example:
  • Sum $ S = 5 $, Product $ P = 6 $:
    $$ x^2 - 5x + 6 = 0 $$

4. Common Concepts

• Quadratic Formula:
  $$ x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
• Vertex of Parabola:
  • Coordinates: $ \left( - \dfrac{b}{2a}, f\left(- \dfrac{b}{2a}\right) \right) $.
• Graph: A parabola opens upwards if $ a > 0 $, downwards if $ a < 0 $.

5. Special Cases

• Equation with No Real Roots: $ x^2 + 1 = 0 $, roots $ x = \pm i $.
• Equation with One Real Root: $ x^2 + 2x + 1 = 0 $, root $ x = -1 $.
• Equation with All Real Roots: $ x^2 - 5x + 6 = 0 $, roots $ x = 2, 3 $.