Class 10 Mathematics
Quadratic Equations
Exercise 4.2 – Stepwise Solutions
1. Find the roots by factorisation
(i) x² − 3x − 10 = 0
x² − 5x + 2x − 10 = 0
x(x − 5) + 2(x − 5) = 0
(x − 5)(x + 2) = 0
Roots: x = 5, −2
(ii) 2x² + x − 6 = 0
2x² + 4x − 3x − 6 = 0
2x(x + 2) − 3(x + 2) = 0
(x + 2)(2x − 3) = 0
Roots: x = −2, 3/2
(iii) √2 x² + 7x + 5√2 = 0
√2 x² + 5x + 2x + 5√2 = 0
x(√2 x + 5) + √2(√2 x + 5) = 0
(√2 x + 5)(x + √2) = 0
Roots: x = −5/√2, −√2
(iv) 2x² − x + 1/8 = 0
Multiply entire equation by 8:
16x² − 8x + 1 = 0
(4x − 1)² = 0
Roots: x = 1/4 (equal roots)
(v) 100x² − 20x + 1 = 0
(10x − 1)² = 0
Roots: x = 1/10 (equal roots)
3. Find two numbers whose sum is 27 and product is 182
Let one number = x
Other number = 27 − x
x(27 − x) = 182
27x − x² = 182
x² − 27x + 182 = 0
(x − 13)(x − 14) = 0
Numbers: 13 and 14
4. Find two consecutive positive integers whose squares sum to 365
Let first integer = x
Second integer = x + 1
x² + (x + 1)² = 365
x² + x² + 2x + 1 = 365
2x² + 2x − 364 = 0
x² + x − 182 = 0
(x − 13)(x + 14) = 0
Integers: 13 and 14
5. Right triangle problem
Let base = x cm
Altitude = x − 7 cm
Using Pythagoras :
x² + (x − 7)² = 13²
x² + x² − 14x + 49 = 169
2x² − 14x − 120 = 0
x² − 7x − 60 = 0
(x − 12)(x + 5) = 0
Base = 12 cm, Altitude = 5 cm
6. Cottage industry problem
Let number of articles = x
Cost per article = 2x + 3
Total cost = x(2x + 3)
x(2x + 3) = 90
2x² + 3x − 90 = 0
2x² + 15x − 12x − 90 = 0
(x − 6)(2x + 15) = 0
Number of articles = 6
Cost per article = ₹15