Exercise 4.1

Similar to Q1(i)

Check if quadratic: (x + 2)² = 3(x - 4)

View Step-by-Step Solution

Step 1: Simplify

LHS: (x + 2)² = x² + 4x + 4
RHS: 3(x - 4) = 3x - 12

Step 2: Equate to Zero

x² + 4x + 4 = 3x - 12
x² + 4x - 3x + 4 + 12 = 0
x² + x + 16 = 0

Yes, it is a Quadratic Equation.

Similar to Q1(ii)

Check if quadratic: x² - 4x = (-3)(4 - x)

View Step-by-Step Solution

Step 1: Simplify

LHS: x² - 4x
RHS: (-3)(4) - (-3)(x) = -12 + 3x

Step 2: Equate to Zero

x² - 4x - 3x + 12 = 0
x² - 7x + 12 = 0

Yes, it is a Quadratic Equation.

Similar to Q1(iii)

Check if quadratic: (x - 3)(x + 2) = (x + 1)(x - 2)

View Step-by-Step Solution

Step 1: Expand Both Sides

LHS: x² + 2x - 3x - 6 = x² - x - 6
RHS: x² - 2x + 1x - 2 = x² - x - 2

Step 2: Simplify

x² - x - 6 = x² - x - 2
Subtract x² from both sides -> terms cancel.
-6 = -2 (False statement, no x² term left)

No, it is NOT a Quadratic Equation (Linear).

Similar to Q1(iv)

Check if quadratic: (x - 2)(3x + 1) = x(x + 3)

View Step-by-Step Solution

Step 1: Expand

LHS: 3x² + x - 6x - 2 = 3x² - 5x - 2
RHS: x² + 3x

Step 2: Rearrange

3x² - 5x - 2 - x² - 3x = 0
2x² - 8x - 2 = 0

Yes, it is a Quadratic Equation.

Similar to Q1(v)

Check if quadratic: (3x - 1)(x - 2) = (x + 4)(x - 1)

View Step-by-Step Solution

Step 1: Expand

LHS: 3x² - 6x - x + 2 = 3x² - 7x + 2
RHS: x² - x + 4x - 4 = x² + 3x - 4

Step 2: Rearrange

(3x² - x²) + (-7x - 3x) + (2 + 4) = 0
2x² - 10x + 6 = 0

Yes, it is a Quadratic Equation.

Similar to Q1(vi)

Check if quadratic: x² + 5x - 2 = (x - 3)²

View Step-by-Step Solution

Step 1: Expand RHS

RHS: (x - 3)² = x² - 6x + 9

Step 2: Compare

x² + 5x - 2 = x² - 6x + 9
The x² terms cancel out.
11x - 11 = 0

No, it is NOT a Quadratic Equation (Linear).

Similar to Q1(vii)

Check if quadratic: (x + 1)³ = x(x² + 5)

View Step-by-Step Solution

Step 1: Expand Cubes

LHS: x³ + 3x²(1) + 3x(1)² + 1³ = x³ + 3x² + 3x + 1
RHS: x³ + 5x

Step 2: Simplify

x³ + 3x² + 3x + 1 = x³ + 5x
Subtract x³ from both sides.
3x² + 3x - 5x + 1 = 0 → 3x² - 2x + 1 = 0

Yes, it is a Quadratic Equation.

Similar to Q1(viii)

Check if quadratic: x³ - 2x² + 5 = (x + 1)³

View Step-by-Step Solution

Step 1: Expand RHS

RHS: (x + 1)³ = x³ + 3x² + 3x + 1

Step 2: Compare

x³ - 2x² + 5 = x³ + 3x² + 3x + 1
Subtract x³ from both sides.
-2x² - 3x² - 3x + 5 - 1 = 0 → -5x² - 3x + 4 = 0

Yes, it is a Quadratic Equation.

Similar to Q2(i)

The area of a rectangular plot is 800 m². The length of the plot (in metres) is two more than thrice its breadth. Find the length and breadth.

View Step-by-Step Solution

Step 1: Define Variables

Let Breadth = x metres.
Length = (3x + 2) metres.

Step 2: Form Equation

Area = Length × Breadth = 800
x(3x + 2) = 800
3x² + 2x - 800 = 0

Equation: 3x² + 2x - 800 = 0

Similar to Q2(ii)

The product of two consecutive positive integers is 210. We need to find the integers.

View Step-by-Step Solution

Step 1: Define Integers

Let first integer = x.
Second integer = x + 1.

Step 2: Form Equation

Product = 210
x(x + 1) = 210
x² + x - 210 = 0

Equation: x² + x - 210 = 0

Similar to Q2(iii)

A mother is 24 years older than her son. The product of their ages (in years) 4 years from now will be 480. We would like to find the son's present age.

View Step-by-Step Solution

Step 1: Current Ages

Son's age = x
Mother's age = x + 24

Step 2: Ages after 4 years

Son = x + 4
Mother = (x + 24) + 4 = x + 28

Step 3: Form Equation

Product = 480
(x + 4)(x + 28) = 480
x² + 28x + 4x + 112 = 480
x² + 32x + 112 - 480 = 0
x² + 32x - 368 = 0

Equation: x² + 32x - 368 = 0

Similar to Q2(iv)

A train travels a distance of 360 km at a uniform speed. If the speed had been 5 km/h more, then it would have taken 1 hour less to cover the same distance. Find the speed of the train.

View Step-by-Step Solution

Step 1: Define Variables

Distance = 360 km
Let original speed = x km/h.
Original time = 360/x hours.

Step 2: New Conditions

New speed = (x + 5) km/h.
New time = 360/(x + 5) hours.

Step 3: Form Equation

Time Difference = 1 hour
(Original Time) - (New Time) = 1
360/x - 360/(x + 5) = 1
360 [ (x + 5 - x) / x(x + 5) ] = 1
360(5) = x² + 5x
1800 = x² + 5x
x² + 5x - 1800 = 0

Equation: x² + 5x - 1800 = 0

Q1. Check if Quadratic

Q2. Formulate Equations