Quadratic Equations — Class 10 (Exam-focused Notes)

Quadratic formula • Discriminant • Factorisation • Completing the square • Graphs • Practice
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1. What is a Quadratic Equation?

A quadratic equation in x is a polynomial equation of degree 2, written in standard form:

ax² + bx + c = 0, where a ≠ 0

Here a, b, c are real coefficients. Solutions (roots) are values of x that satisfy the equation.

2. Methods to Solve a Quadratic

  • Quadratic Formula: x = [−b ± √(b² − 4ac)] / (2a)
  • Factorisation: Express ax² + bx + c as (px + q)(rx + s) and find roots.
  • Completing the Square: Convert to (x + d)² = e form and solve.
  • Graphically: Intersection points of y = ax² + bx + c with x-axis.

3. Discriminant & Nature of Roots

The discriminant Δ = b² − 4ac determines the type of roots:

ΔType of RootsRemarks
Δ > 0Two distinct real rootsQuadratic crosses x-axis twice
Δ = 0One repeated (real) rootTangent to x-axis at vertex
Δ < 0Two complex (non-real) rootsNo real x-intercepts

Quick check: For x² − 4x + 3: Δ = 16 − 12 = 4 > 0 ⇒ two real roots (x=1,3).

4. Completing the Square (Short recipe)

To solve ax² + bx + c = 0 (with a ≠ 0): divide by a → x² + (b/a)x + c/a = 0. Then add & subtract (b/2a)²:

x² + (b/a)x + (b/2a)² = (b² − 4ac) / (4a²) ⇒ (x + b/2a)² = (b² − 4ac) / (4a²)

Take square roots and rearrange to reach quadratic formula.

5. Graphs & Vertex Form

Parabola y = ax² + bx + c. Vertex at x = −b/(2a). Vertex form: y = a(x − h)² + k where (h,k) is vertex.

  • If a > 0: parabola opens upwards (minimum at vertex).
  • If a < 0: parabola opens downwards (maximum at vertex).
Vertex coordinates: (−b/(2a), f(−b/(2a))).

6. Solved Examples (Exam-style)

Example 1. Solve x² − 5x + 6 = 0.

Factor: (x − 2)(x − 3) = 0 ⇒ x = 2, 3.

Example 2. Solve 2x² − 3x − 5 = 0 using quadratic formula.

Δ = 9 + 40 = 49 ⇒ x = [3 ± 7]/4 ⇒ x = (3+7)/4 = 10/4 = 5/2 or x = (3−7)/4 = −1.

Example 3. Complete the square: x² + 6x + 5 = 0 ⇒ (x + 3)² − 4 = 0 ⇒ (x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = −1, −5.

7. Practice Questions

  1. Solve x² + 4x + 3 = 0. (Answer toggle)
  2. Find nature of roots for 3x² + 2x + 5 = 0.
  3. For which value of k will the equation x² + kx + 9 = 0 have equal roots?
  4. Find vertex of y = 2x² − 8x + 6.

8. Exam Tips

  • Check discriminant first to decide method: if Δ is a perfect square, use formula/factorisation; if Δ = 0, answer is repeated root.
  • Use completing the square to find vertex quickly.
  • Practice factoring quadratics with leading coefficient ≠ 1 by splitting middle term or AC method.