| --- |
| source: ncert |
| topic: algebra |
| class_level: class_10 |
| chapter: ch4 |
| difficulty: intermediate |
| --- |
| |
| Class 10 | Ch4: Quadratic Equations |
| STANDARD FORM: ax^2+bx+c=0 where a≠0. |
| METHODS OF SOLVING: |
| Method 1 - Factorisation: |
| Split middle term: find p,q such that p+q=b and p×q=a×c. |
| 2x^2-7x+3=0. a×c=6. p+q=-7, p×q=6. p=-6,q=-1. |
| 2x^2-6x-x+3=0 → 2x(x-3)-1(x-3)=0 → (2x-1)(x-3)=0. |
| x=1/2 or x=3. |
| Method 2 - Completing the Square: |
| ax^2+bx+c=0 → x^2+(b/a)x=-c/a → add (b/2a)^2 both sides. |
| (x+b/2a)^2=(b^2-4ac)/4a^2 → x=-b/2a ± sqrt(b^2-4ac)/2a. |
| Method 3 - Quadratic Formula: |
| x=(-b ± sqrt(b^2-4ac))/2a. (Derived from completing the square.) |
| DISCRIMINANT D=b^2-4ac: |
| D>0: two distinct real roots. D=0: two equal real roots (repeated). D<0: no real roots (complex). |
| NATURE OF ROOTS: |
| If D is perfect square: roots are rational. If D>0 but not perfect square: roots irrational. |
| RELATIONSHIP BETWEEN ROOTS: |
| Sum of roots=alpha+beta=-b/a. Product of roots=alpha×beta=c/a. |
| WORD PROBLEMS - STANDARD TYPES: |
| Number problems, age problems, speed-distance problems, area problems. |
| SOLVED EXAMPLES: |
| Example 1: 2x^2-7x+3=0 by factorisation. |
| a×c=6. Factors of 6 summing to -7: -6 and -1. |
| 2x^2-6x-x+3=2x(x-3)-1(x-3)=(2x-1)(x-3)=0. |
| x=1/2 or x=3. |
| Example 2: Find nature of roots of 2x^2-6x+3=0. |
| D=36-24=12>0. Two distinct real roots (D not perfect square, so irrational). |
| Example 3: Sum of ages of father and son is 45. Five years ago product was 124. |
| Let son's age=x. Father's age=45-x. |
| (x-5)(45-x-5)=124 → (x-5)(40-x)=124. |
| 40x-x^2-200+5x=124 → x^2-45x+324=0 → (x-9)(x-36)=0. |
| x=9 (son) or x=36 (check: son can't be 36 if sum is 45). Son=9, Father=36. |
| COMMON MISTAKES: |
| Discriminant: D=b^2-4ac NOT b^2+4ac. |
| If D=0: ONE repeated root (two equal roots, not no roots). |
| Quadratic formula: entire (-b±sqrt(D)) is over 2a. |
