📐 MATHS · CLASS 10

Quadratic Equations

Parabolas, roots and the magical quadratic formula — all animated

1 / 6

📌 What is a Quadratic Equation?

A polynomial equation of degree 2. The highest power of x is 2.

Standard Form ax² + bx + c = 0   (where a ≠ 0)
  • a, b, c are real numbers — called coefficients
  • Example: 2x² − 5x + 3 = 0  →  a=2, b=−5, c=3
  • Example: x² − 4 = 0  →  a=1, b=0, c=−4
  • The graph of ax² + bx + c is a parabola
2 / 6

📈 The Parabola — Graph of y = x² − 5x + 6

The parabola crosses the x-axis at the roots (zeros) of the equation. Watch it draw!

Roots (where y = 0):   x = 2   and   x = 3
3 / 6

🛠️ Methods to Solve Quadratic Equations

1

Factorisation

Split the middle term and factorise. Works when roots are rational numbers.

2

Completing the Square

Convert to (x + p)² = q form and take square root of both sides.

3

Quadratic Formula

Works for ALL quadratics — even when factorisation is hard.

4 / 6

⭐ The Quadratic Formula

Derived by completing the square on ax² + bx + c = 0:

🌟 Quadratic Formula x = [−b ± √(b² − 4ac)] / 2a

Discriminant (D) = b² − 4ac tells us the nature of roots:

📈📈

D > 0

Two distinct real roots — parabola crosses x-axis at 2 points

📍

D = 0

Two equal (repeated) real roots — parabola touches x-axis at 1 point

🚫

D < 0

No real roots — parabola doesn't cross the x-axis

5 / 6

✏️ Solved Example

Q: Solve 2x² − 7x + 3 = 0 using the quadratic formula

1

Identify a, b, c

a = 2, b = −7, c = 3

2

Calculate discriminant D

D = (−7)² − 4×2×3 = 49 − 24 = 25

3

Apply formula

x = (7 ± √25) / (2×2) = (7 ± 5) / 4

4

Find both roots

x₁ = (7 + 5)/4 = 3    x₂ = (7 − 5)/4 = ½

Roots: x = 3   and   x = 0.5
6 / 6

🧠 Quick Quiz

Q: For x² − 5x + 6 = 0, what is the discriminant?

Excellent! Quadratic Equations mastered 🎉

← AP Lesson