Exercise 4.1

(i)

(i) x+12 = 2(x - 3)

Solution

Since a+b2 = a2 + b2 +

x2 + + = (2x - 6)

x2 + 2x + 1 - (2x - 6) =

x2 + 2x + 1 - + = 0

x2 + = 0

Here, the degree of x2 + 7 = 0 is .

∴ It is a quadratic equation.

(ii)

(ii) x2 - 2x = (-2) (3 - x)

Solution

x2 - 2x = - + x

x2 - 2x - + =

x2 - + 6 = 0

Degree =

∴ It is a quadratic equation.

(iii)

(iii) (x - 2)( x + 1) = (x -1)( x + 3)

Solution

x2 - + x - = x2 + -

x2 - x - = x2 + 2x

x2 - x - 2 - x2 - + = 0

- + = 0

Degree =

∴ It is not a quadratic equation.

(iv)

(iv) (x - 3)(2x +1) = x ( x + 5)

Solution

2x2 + x - - = x2 + 5x

2x2 - - 3 = x2 + 5x

2x2 - 5x - 3 -x2 - =

x2 - - = 0

Degree =

∴ It is a quadratic equation.

(v)

(v) (2x -1)(x - 3) = (x + 5)(x -1)

Solution

2x2 - 6x - + 3 = x2 - x + 5x -

2x2 - x + 3 = x2 + x - 5

2x2 - 7x + 3 - x2 - 4x + =

x2 - x + = 0

Degree =

∴ It is a quadratic equation.

(vi)

(vi) x2 + 3x +1 = x−22

Solution

x2 + 3x + 1 = x2 - x + [Note : a−b2 = a2 - 2ab + b2]

x2 + 3x + 1 - x2 + 4x - =

x - = 0

Degree =

∴ It is not a quadratic equation.

(vii)

(vii) x+23 = 2x (x2 -1)

Solution

We know that, a+b3 = a3a2 + 3a2b + 3ab2 + b3b2

x3 + 3x2 (2) + 3(x)22 + 23 = 2x3 - x

x3 + x2 + x + = 2x3 - x

- x3 + 6x2 + x + = 0

Degree =

∴ It is not a quadratic equation.

(viii)

(viii) x3 - 4x2 - x + 1 = x−23

Solution

x3 - 4x2 - x + 1 = x3x2 - 3x2(2) + 3(x)22 - 23 [Note∵ a−b3 = a3 - 3a2b + 3ab2 - b3]

x3 - 4x2 - x + 1 =x3 - x2 + x -

x3 - 4x2 - x + 1 - x3 + 6x2 - 12x + =

2x2 - x + = 0

Degree =

∴ It is a quadratic equation.

(i)

The area of a rectangular plot is 528 m2. The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot.

Solution

We know that the area of a rectangle can be expressed as the product of its length and breadth.

Since we don’t know the length and breadth of the given rectangle, we assume the breadth of the plot to be a variable (x meters). Then, we use the given relationship between length and breadth: length = 1 + 2 times breadth.

Therefore, Length = 2x +

Area of rectangle = Length × Breadth

Breadth = x

Length = 2x +

Area of Rectangular Plot = Length × Breadth = m2

(2x + 1) × (x) =

x2 + x = 528

2x2 + x - =

Thus, the quadratic equation is 2x2 + x - 528 = 0 , where x is the breadth of the rectangular plot.

(ii)

The product of two consecutive positive integers is 306. We need to find the integers.

Solution

Let the first integer be x.

Since the integers are consecutive, the next integer is (x + 1).

It is given that

First integer × Next integer =

Therefore,

x (x + 1) =

x2x3 + = 306

x2 + x - =

Thus, the quadratic equation is x2 + x - 306 = 0 where x is the integer.

(iii)

Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

Solution

Let's assume that Rohan’s present age is x years.

Then, from the first condition, his mother’s age is (x + ) years.

Three years from now, Rohan’s age will be (x + 3) and Rohan’s mother age will be (x + 3) + 26 = (x + ). The product of their ages is 360.

Therefore,

(x + 3) × (x + 29) =

x2 + x + 3x + = 360

x2 + x + 87 - =

x2 + 32x - = 0

Thus, the quadratic equation is x2 + 32x - 273 = 0 where x is the present age of Rohan.

(iv)

A train travels a distance of 480 km at a uniform speed. If the speed had been 8 kmh less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Solution

Distance is equal to speed multiplied by time. Let the speed be s kmh and time be t hours.

D = st

= st …(i)

t = 480ss480 …(ii)

As per the given conditions, for the same distance covered at a speed reduced by 8 kmh, the time taken would have increased by hours.

Therefore,

(s - 8)(t + 3) =

st + s - 8t - = 480

480 + 3s - (480s) - = 480 [From equations (i) and (ii)]

3s - 3840s - 24 =

3s (s) - 3840 - 24 (s) = 0

3s2 - 24s - = 0

(3 s2 - 24s - 3840)/3 = 0

s2 ­- 8s - = 0

Thus, the quadratic equation is s2 - 8s - 1280 = 0, where s is the speed of the train.