Sunday, December 16, 2018

Quadratic Equations Exercise 4.1

Quadratic Equations Exercise 4.1

 

Page No: 73

1. Check whether the following are quadratic equations:
(i) (x + 1)2 = 2(x - 3)
(ii) x2 - 2x = (-2)(3 - x)
(iii) (x - 2)(x +1) = (x - 1)(x + 3)
(iv) (x - 3)(2x + 1) = x(x + 5)
(v) (2x - 1)(x - 3) = (x + 5)(x - 1)
(vi) x2 + 3x + 1 = (x - 2)2
(vii) (x + 2)3 = 2x(x2 - 1)
(viii) x3 - 4x2 - x + 1 = (x - 2)3

Answer

(i) (x + 2)2 = 2(x - 3)
⇒ x2 + 2x + 1 = 2x - 6
⇒ x2 + 7 = 0
It is of the form ax2 + bx + c = 0.
Hence, the given equation is quadratic equation.

(ii) x2 - 2x = (-2)(3 - x)
⇒ x- 2x = -6 + 2
⇒ x- 4x + 6 = 0
It is of the form ax2 + bx + c = 0.
Hence, the given equation is quadratic equation.

(iii) (x - 2)(x + 1) = (x - 1)(x + 3)
⇒ xx - 2 = x+ 2x - 3
⇒ 3x - 1 =0
It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.

(iv) (x - 3)(2x + 1) = x(x + 5)
⇒ 2x- 5x - 3 = x+ 5x
⇒  x- 10x - 3 = 0
It is of the form ax2 + bx + c = 0.
Hence, the given equation is quadratic equation.

(v) (2x - 1)(x - 3) = (x + 5)(x - 1)
⇒ 2x- 7x + 3 = x+ 4- 5
⇒ x- 11x + 8 = 0
It is of the form ax2 + bx + c = 0.
Hence, the given equation is quadratic equation.

(vi) x2 + 3x + 1 = (x - 2)2
⇒ x2 + 3x + 1 = x2 + 4 - 4x
⇒ 7x - 3 = 0
It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.

(vii) (x + 2)3 = 2x(x2 - 1)
⇒ x3 + 8 + x2 + 12x = 2x3 - 2x
⇒ x3 + 14x - 6x2 - 8 = 0
It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.

(viii) x3 - 4x2 - x + 1 = (x - 2)3
⇒  x3 - 4x2 - x + 1 = x3 - 8 - 6x + 12x
⇒ 2x2 - 13x + 9 = 0
It is of the form ax2 + bx + c = 0.
Hence, the given equation is 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 (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Answer

Let the breadth of the rectangular plot = x m
Hence, the length of the plot is (2x + 1) m.
Formula of area of rectangle = length × breadth = 528 m2
Putting the value of length and width, we get
(2+ 1) × x = 528
⇒ 2x2 + x =528
⇒ 2x2 + x  - 528 = 0

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

Answer

Let the first integer number = x
Next consecutive positive integer will =  x + 1
Product of both integers = x × (x +1) = 306
⇒ xx = 306
⇒ xx - 306 = 0

(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.

Answer

Let take Rohan's age = x years
Hence, his mother's age = x + 26
3 years from now
Rohan's age = x + 3
Age of Rohan's mother will = x + 26 + 3 = x + 29
The product of their ages 3 years from now will be 360 so that
(x + 3)(x + 29) = 360
⇒ x2 + 29x + 3x + 87 = 360
⇒ x2 + 32x + 87 - 360 = 0
⇒ x2 + 32x - 273  = 0

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

Answer

Let the speed of train be x km/h.
Time taken to travel 480 km = 480/x km/h

In second condition, let the speed of train = (x - 8) km/h
It is also given that the train will take 3 hours to cover the same distance.
Therefore, time taken to travel 480 km = (480/x + 3) km/h

Speed × Time = Distance

(x - 8)(480/x + 3) = 480
⇒ 480 + 3x - 3840/x - 24 = 480
⇒ 3x - 3840/x = 24
⇒ 3x- 8x - 1280 = 0

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