Class 10 Mathematics
Quadratic Equations
Exercise 4.1 – Stepwise Solutions
1. Check whether the following are quadratic equations
(i) (x + 1)2 = 2(x − 3)
Step 1: Expand both sides
x² + 2x + 1 = 2x − 6
Step 2: Bring all terms to one side
x² + 2x + 1 − 2x + 6 = 0
x² + 7 = 0
It is a quadratic equation.
(ii) x² − 2x = (−2)(3 − x)
x² − 2x = −6 + 2x
x² − 2x + 6 − 2x = 0
x² − 4x + 6 = 0
It is a quadratic equation.
(iii) (x − 2)(x + 1) = (x − 1)(x + 3)
x² − x − 2 = x² + 2x − 3
x² − x − 2 − x² − 2x + 3 = 0
−3x + 1 = 0
It is not quadratic (linear equation).
(iv) (x − 3)(2x + 1) = x(x + 5)
2x² − 5x − 3 = x² + 5x
2x² − 5x − 3 − x² − 5x = 0
x² − 10x − 3 = 0
It is a quadratic equation.
(v) (2x − 1)(x − 3) = (x + 5)(x − 1)
2x² − 7x + 3 = x² + 4x − 5
2x² − 7x + 3 − x² − 4x + 5 = 0
x² − 11x + 8 = 0
It is a quadratic equation.
(vi) x² + 3x + 1 = (x − 2)2
x² + 3x + 1 = x² − 4x + 4
x² + 3x + 1 − x² + 4x − 4 = 0
7x − 3 = 0
It is not quadratic (linear equation).
(vii) (x + 2)3 = 2x(x² − 1)
x³ + 6x² + 12x + 8 = 2x³ − 2x
x³ + 6x² + 12x + 8 − 2x³ + 2x = 0
−x³ + 6x² + 14x + 8 = 0
It is not quadratic (cubic equation).
(viii) x³ − 4x² − x + 1 = (x − 2)3
x³ − 4x² − x + 1 = x³ − 6x² + 12x − 8
x³ − 4x² − x + 1 − x³ + 6x² − 12x + 8 = 0
2x² − 13x + 9 = 0
It is a quadratic equation.
2. Represent the following situation in the form of a quadratic equation
(i) Area of a rectangular plot = 528 m²
Let breadth = x metres
Length = 2x + 1 metres
Area = Length × Breadth
x(2x + 1) = 528
2x² + x = 528
2x² + x − 528 = 0
Required quadratic equation: 2x² + x − 528 = 0