knowledge-base/class_10/ch2/0163_algebra.md

---
source: ncert_exercises
topic: algebra
class_level: class_10
chapter: ch2
exercise: ex2.2
difficulty: intermediate
---

Class 10 | Ch2: Polynomials | Exercise 2.2

Key Formulas for quadratic polynomial ax²+bx+c with zeroes α and β:
Sum of zeroes: α+β = -b/a
Product of zeroes: αβ = c/a
Quadratic polynomial with given zeroes: x²-(α+β)x+αβ

Q1. Find zeroes of the following quadratic polynomials and verify relationship between zeroes and coefficients:
(i) x²-2x-8  (ii) 4s²-4s+1  (iii) 6x²-3-7x  (iv) 4u²+8u  (v) t²-15  (vi) 3x²-x-4
Answer:
(i) x²-2x-8=(x-4)(x+2). Zeroes: 4,-2. Sum=4+(-2)=2=2/1=−(−2)/1 ✓. Product=4×(−2)=−8=−8/1 ✓. ✅
(ii) 4s²-4s+1=(2s-1)². Zeroes: 1/2,1/2. Sum=1=4/4 ✓. Product=1/4=1/4 ✓. ✅
(iii) 6x²-7x-3=(3x+1)(2x-3). Zeroes: -1/3, 3/2. Sum=-1/3+3/2=7/6=7/6 ✓. Product=-1/3×3/2=-1/2=-3/6 ✓. ✅
(iv) 4u²+8u=4u(u+2). Zeroes: 0,-2. Sum=-2=-8/4 ✓. Product=0=0/4 ✓. ✅
(v) t²-15=(t-√15)(t+√15). Zeroes: √15,-√15. Sum=0 ✓. Product=-15 ✓. ✅
(vi) 3x²-x-4=(3x-4)(x+1). Zeroes: 4/3,-1. Sum=4/3-1=1/3=1/3 ✓. Product=-4/3=-4/3 ✓. ✅

Q2. Find a quadratic polynomial whose sum and product of zeroes are given:
(i) 1/4, -1  (ii) √2, 1/3  (iii) 0, √5  (iv) 1, 1  (v) -1/4, 1/4  (vi) 4, 1
Answer: Polynomial = k[x²-(sum)x+product].
(i) x²-x/4-1 → multiply by 4: 4x²-x-4. ✅
(ii) x²-√2x+1/3 → 3x²-3√2x+1. ✅
(iii) x²-0x+√5=x²+√5. ✅
(iv) x²-x+1. ✅
(v) x²+x/4+1/4 → 4x²+x+1. ✅
(vi) x²-4x+1. ✅