### Problem On Quadratic Equations

**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})