📚 Concepts

📚 Quadratic Equation  

A polynomial equation of the form: 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0, where 𝑎 ≠ 0

📚 Degree of the Equation  

The highest exponent of the variable is 2 → Quadratic

📚 Roots or Zeros  

Values of 𝑥 for which the equation becomes true (i.e., LHS = RHS = 0)

📚 Number of Roots  

A quadratic equation always has 2 roots (real or complex)

📚 Methods to Solve Quadratic Equations  

🟢 Factorisation – Splitting the middle term  

🟡 Completing the Square – Making a perfect square  

🔴 Quadratic Formula – Using 𝑥 = (−𝑏 ± √(𝑏² − 4𝑎𝑐)) ÷ 2𝑎

📚 Verification of Roots  

Substitute each root into the original equation to check if it becomes 0

📚 Perfect Square Condition  

If 𝑏² = 4𝑎𝑐, then the quadratic is a perfect square trinomial

✍️ Formulas

✍️ Standard Form of a Quadratic Equation:  

𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0, where 𝑎 ≠ 0

✍️ Quadratic Formula:  

𝑥 = (−𝑏 ± √(𝑏² − 4𝑎𝑐)) ⁄ 2𝑎

✍️ Discriminant (𝐷):  

𝐷 = 𝑏² − 4𝑎𝑐

👉 Nature of Roots based on 𝐷:  

🔹 𝐷 > 0 → Two distinct real roots  

🔹 𝐷 = 0 → Two equal real roots  

🔹 𝐷 < 0 → No real roots (complex roots)

✍️ Sum and Product of Roots (𝛼 and 𝛽):  

• 𝛼 + 𝛽 = −𝑏 ⁄ 𝑎   

• 𝛼 × 𝛽 = 𝑐 ⁄ 𝑎

✍️ Factoring Method (when possible):  

Split the middle term such that:    

• Product = 𝑎 × 𝑐    

• Sum = 𝑏

✍️ Completing the Square (Steps):  

• Bring the constant term to RHS  

• Divide entire equation by 𝑎  

• Add (𝑏 ⁄ 2𝑎)² to both sides  

• Convert LHS into a perfect square  

• Solve by taking square roots on both sides