A quadratic equation is a polynomial of degree 2 in the variable x:

ax² + bx + c = 0   where a, b, c ∈ ℚ, a ≠ 0.

Terms: a — coefficient of x², b — coefficient of x, c — constant term.

2. Methods of Solving

A. Factorization

1. Write ax2 + bx + c = 0
2. Factor into linear terms (if possible)
3. Use zero-product property

B. Square Root Method

If the equation can be written x2 = k, then x = ±√k.

C. Completing the Square

Rewrite to form a perfect square: (x + b/(2a))2 = (b2 - 4ac)/(4a2), then solve.

D. Quadratic Formula

For ax2 + bx + c = 0:

x = (-b ± sqrt(b*b - 4*a*c)) / (2*a)

3. Nature of Roots

The discriminant D = b2 - 4ac determines the nature:

D	Nature	Example
D > 0	Two distinct real roots	x2 - 5x + 6 → 2, 3
D = 0	Two equal real roots	x2 - 4x + 4 → 2, 2
D < 0	No real roots (complex)	x2 + x + 1

4. Relationship Between Roots and Coefficients

If the roots are α and β for ax2 + bx + c = 0 then:

• α + β = -b/a
• αβ = c/a

5. Forming Quadratics from Roots

Given roots α and β, an equation with leading coefficient 1 is:

x2 - (α + β)x + (αβ) = 0

6. Word Problems (brief)

• Area. Area of a square = 49 ⇒ side x: x2=49 ⇒ x=7 (positive).
• Motion. If a train takes 2 hours less at 5 km/h faster — leads to a quadratic in speed. (Set up: distances equal.)
• Mixture / Numbers. Two numbers with sum 10 and product 21 ⇒ equation x2 - 10x + 21 = 0.

7. Quick Summary Table

Concept	Formula / Method	Example
Standard Form	ax2+bx+c=0	2x2+5x-3=0
Factorization	Factor & set each factor = 0	x2-5x+6=(x-2)(x-3)
Square Roots	x2=k → x=±sqrt(k)	x2=16 → x=±4
Completing Square	(x+b/2a)2=...	x2+6x+5→x=-1,-5
Quadratic Formula	x=(-b±sqrt(b2-4ac))/(2a)	2x2-4x-6 → 3,-1

8. Exercises

1. Solve by factorization: x2 - 3x - 10 = 0
2. Solve by quadratic formula: x2 + 5x + 6 = 0
3. Solve by completing the square: x2 + 6x + 5 = 0
4. Find nature of roots: 2x2 - 4x + 2 = 0
5. Form a quadratic equation whose roots are 4 and -3.