Algebraic Equations 📚✨

Algebraic equations are polynomial equations used in quantitative aptitude exams. They can be of degree 1 (linear), degree 2 (quadratic), or degree 3 (cubic)¹.

Linear Equations (Degree 1) ➗

• Form: $ ax + c = 0 $
• Examples:
  • $ 2x + 3y = 4 $
  • $ x + y + z = 10 $
• Solving Linear Equations:
  • Use substitution or elimination to find variable values.
  • Example:
    • $ 2x + 3y = 13 $ …(1)
    • $ 3x + 2y = 12 $ …(2)
    • Multiply and subtract to eliminate a variable:
      • $ 3 \times (2) - 2 \times (1) $ → $ 5x = 10 $ → $ x = 2 $
      • Substitute $ x $ into (1): $ y = 3 $
    • Result: $ x < y $ ✅
  • Another Example:
    • $ 4x + 5y = 14 $ …(1)
    • $ 2x + 3y = 5 $ …(2)
    • Multiply (2) by 2: $ 4x + 6y = 10 $
    • Subtract (1): $ y = -4 $, then $ x = 1 $
    • Result: $ x > y $ 👍

Quadratic Equations (Degree 2) 🟪

• Form: $ ax^2 + bx + c = 0 $
• Examples:
  • $ x^2 + 2x + 3 = 0 $
  • $ y^2 - 3y + 4 = 0 $

Solving Quadratic Equations:

1. Factorization Method 🧩
   • Split the middle term and factorize.
   • Example:
     • $ x^2 - 2x - 15 = 0 $
     • $ x^2 - 5x + 3x - 15 = 0 $
     • $ x(x-5) + 3(x-5) = 0 $
     • $ (x+3)(x-5) = 0 $
     • Solutions: $ x = -3 $ or $ x = 5 $ 🎯
2. Sridharacharya’s (Quadratic Formula) Method 🧮
   • Formula: $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $
   • Example:
     • For $ x^2 - 2x - 15 = 0 $:
       • $ x_1 = 5 $, $ x_2 = -3 $ ✔️
3. Comparing Roots
   • Example:
     • $ x^2 = 625 $ → $ x = +25 $ or $ -25 $
     • $ y = \sqrt{625} $ → $ y = 25 $
     • Result: $ x \leq y $ 🟰
   • Example:
     • $ m = \sqrt{324} = 18 $
     • $ n^2 - 16n - 36 = 0 $ → $ n = 18 $ or $ n = -2 $
     • Result: $ m \geq n $ 🆚

Cubic Equations (Degree 3) 🟦

• Form: $ ax^3 + bx^2 + cx + d = 0 $
• Examples:
  • $ x^3 + 2x^2 + 3x + 4 = 0 $
  • $ 2x^3 + 12x^2 + 30x + 48 = 0 $

Solving and Comparing:

• Example:
  • $ x = \sqrt{15625} = 25 $
  • $ y^2 = 625 $ → $ y = +25, -25 $
  • Result: $ y \leq x $ 🧑‍🏫

Key Points to Remember 📝

• Linear equations: Degree 1, straight-line solutions.
• Quadratic equations: Degree 2, solved by factorization or quadratic formula.
• Cubic equations: Degree 3, solved by finding cube roots or factorization.
• Always compare solutions as required (e.g., $ x < y $, $ x \geq y $).
• For square roots, only the positive root is considered unless otherwise specified.
• For quadratic equations, always check for both possible roots.

Emojis Legend

• ➗ Linear Equations
• 🟪 Quadratic Equations
• 🟦 Cubic Equations
• 🧩 Factorization
• 🧮 Formula Method
• 🎯 Solution
• 🆚 Comparison
• 🟰 Equal/Relation
• 🧑‍🏫 Explanation

You can use these notes to quickly revise and practice algebraic equations for quant aptitude tests!¹