Class 10 Mathematics
Quadratic Equations
Exercise 4.3 – Stepwise Solutions
1. Find the nature of the roots. If real roots exist, find them.
(i) 2x² − 3x + 5 = 0
a = 2, b = −3, c = 5
D = b² − 4ac
D = (−3)² − 4(2)(5)
D = 9 − 40 = −31
Since D < 0, the roots are imaginary (no real roots).
(ii) 3x² − 4√3 x + 4 = 0
a = 3, b = −4√3, c = 4
D = (−4√3)² − 4(3)(4)
D = 48 − 48 = 0
Since D = 0, roots are real and equal.
x = −b / 2a
x = 4√3 / 6
x = 2√3 / 3
Equal roots: x = 2√3 / 3
(iii) 2x² − 6x + 3 = 0
a = 2, b = −6, c = 3
D = (−6)² − 4(2)(3)
D = 36 − 24 = 12
Since D > 0, roots are real and distinct.
x = [6 ± √12] / 4
x = [6 ± 2√3] / 4
x = (3 ± √3) / 2
Roots: (3 + √3)/2 and (3 − √3)/2
2. Find the value of k for equal roots.
(i) 2x² + kx + 3 = 0
For equal roots, D = 0
D = k² − 4(2)(3)
D = k² − 24
k² − 24 = 0
k² = 24
k = ±2√6
k = 2√6 or −2√6
(ii) kx(x − 2) + 6 = 0
kx² − 2kx + 6 = 0
a = k, b = −2k, c = 6
D = (−2k)² − 4(k)(6)
D = 4k² − 24k
For equal roots:
4k² − 24k = 0
4k(k − 6) = 0
k = 0 or 6
Since k ≠ 0 (equation must remain quadratic)
k = 6
3. Mango Grove Problem
Let breadth = x m
Length = 2x m
Area = 800 m²
2x² = 800
x² = 400
x = 20
Breadth = 20 m, Length = 40 m
4. Ages Problem
Let ages be x and 20 − x
Four years ago:
(x − 4)(16 − x) = 48
16x − x² − 64 + 4x = 48
−x² + 20x − 64 = 48
x² − 20x + 112 = 0
D = 400 − 448 = −48
Since D < 0, the situation is not possible.
5. Rectangular Park Problem
Perimeter = 80 m
2(L + B) = 80
L + B = 40
Let L = x, then B = 40 − x
x(40 − x) = 400
40x − x² = 400
x² − 40x + 400 = 0
(x − 20)² = 0
Length = 20 m, Breadth = 20 m
Yes, it is possible.