Concepts to Remember for Problem on Quadratic Equations

1. Nature of Roots of a Quadratic Equation:

• Discriminant, (Δ = b^2 - 4ac) determines the nature of roots.
  • (Δ > 0): Two distinct real roots
  • (Δ = 0): Two equal real roots (Repeated roots)
  • (Δ < 0): No real roots (Complex roots)

2. Conditions for Equal Roots (Repeated Roots):

• Discriminant, (Δ = b^2 - 4ac = 0)
• Roots are equal to: (x = \frac{-b \pm \sqrt{Δ}}{2a} = \frac{-b}{2a})

3. Relationship between Roots and Coefficients of a Quadratic Equation:

• Product of roots: (p = \frac{c}{a})
• Sum of roots: (q = \frac{-b}{a})

4. Sum and Product of Roots of a Quadratic Equation:

• Sum of roots: (x_1 + x_2 = q = \frac{-b}{a})
• Product of roots: (x_1 x_2 = p = \frac{c}{a})

5. Equations Reducible to Quadratic Equations:

• Equations that can be transformed into a standard form of a quadratic equation (ax^2 + bx + c = 0) by suitable transformations.

6. Solving Quadratic Equations:

• Factorization: Splitting the quadratic expression into the product of linear factors.
• Completing the square: Transforming the equation into a perfect square form.
• Quadratic Formula: (x = \frac{-b \pm \sqrt{Δ}}{2a}) provides solutions for both real and complex roots.

7. Applications of Quadratic Equations:

• Solving various real-world problems in areas such as geometry, physics, projectile motion, and other applied mathematics.

8. Graphical Representation of Quadratic Equations:

• Sketching parabolas using their vertex form and understanding their properties like minima, maxima, and symmetry.

9. Concept of Discriminant in Determining Types of Roots:

• Discriminant, (Δ = b^2 - 4ac) categorizes the roots as real and distinct, repeated, or complex.

10. Solving Quadratic Equations using Vieta’s Formulas:

• Utilizing Vieta’s formulas when coefficients are in the form (ax^2 + bx + c = 0).
  • Sum of roots: (x_1 + x_2 = -\frac{b}{a})
  • Product of roots: (x_1 x_2 = \frac{c}{a})