Exercise 4.1 Solutions
Quadratic Equations
1. Check whether the following are quadratic equations:
An equation is quadratic if it can be written in the form ax² + bx + c = 0, where a ≠ 0.
(i) (x + 1)² = 2(x − 3)
x2 + 2x + 1 = 2x – 6
x2 + 7 = 0. Yes, it’s a quadratic equation.
(ii) x² − 2x = (−2)(3 − x)
x2 – 2x = -6 + 2x
x2 – 4x + 6 = 0. Yes, it’s a quadratic equation.
(iii) (x − 2)(x + 1) = (x − 1)(x + 3)
x2 + x – 2x – 2 = x2 + 3x – x – 3
x2 – x – 2 = x2 + 2x – 3
-3x + 1 = 0. No, the x² term cancels out.
(iv) (x − 3)(2x + 1) = x(x + 5)
2x2 + x – 6x – 3 = x2 + 5x
2x2 – 5x – 3 = x2 + 5x
x2 – 10x – 3 = 0. Yes, it’s a quadratic equation.
(v) (2x − 1)(x − 3) = (x + 5)(x − 1)
2x2 – 6x – x + 3 = x2 – x + 5x – 5
2x2 – 7x + 3 = x2 + 4x – 5
x2 – 11x + 8 = 0. Yes, it’s a quadratic equation.
(vi) x² + 3x + 1 = (x − 2)²
x2 + 3x + 1 = x2 – 4x + 4
7x – 3 = 0. No, the x² term cancels out.
(vii) (x + 2)³ = 2x(x² − 1)
x3 + 8 + 6x2 + 12x = 2x3 – 2x
x3 – 6x2 – 14x – 8 = 0. No, it’s a cubic equation.
(viii) x³ − 4x² − x + 1 = (x − 2)³
x3 – 4x2 – x + 1 = x3 – 8 – 6x2 + 12x
-4x2 – x + 1 = -6x2 + 12x – 8
2x2 – 13x + 9 = 0. Yes, it’s a quadratic equation.
2. Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528 m². The length of the plot is one more than twice its breadth…
Let breadth be x metres. Then length = 2x + 1 metres.
Area = length × breadth = (2x + 1)x = 528.
2x2 + x = 528
Equation: 2x² + x - 528 = 0
(ii) The product of two consecutive positive integers is 306…
Let the integers be x and x + 1.
Product = x(x + 1) = 306.
x2 + x = 306
Equation: x² + x - 306 = 0
(iii) Rohan’s mother is 26 years older than him…
Let Rohan’s present age be x years. Mother’s age = x + 26.
After 3 years: Rohan’s age = x + 3, Mother’s age = x + 26 + 3 = x + 29.
Product of their ages = (x + 3)(x + 29) = 360.
x2 + 29x + 3x + 87 = 360
x2 + 32x – 273 = 0
Equation: x² + 32x - 273 = 0
(iv) A train travels a distance of 480 km at a uniform speed…
Let the uniform speed be x km/h.
Time taken (t₁) = Distance / Speed = 480/x.
New speed = x - 8 km/h.
New time taken (t₂) = 480/(x-8).
Given: t₂ = t₁ + 3.
480x-8 – 480x = 3
480[ 1x-8 – 1x ] = 3
160[ x – (x-8)x(x-8) ] = 1
160[ 8x2-8x ] = 1
1280 = x2 – 8x
Equation: x² - 8x - 1280 = 0
