Question 2 Represent the following situations in the form of quadratic equations. (i) The area of a rectangular plot is 528 m^2 . 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. (ii) The product of two consecutive positive integers is 306. We need to find the integers. (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. (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.
Class 10 - Math - Quadratic Equations Page 73
Answers
Quadratic equation:
An equation of the form ax²+bx +c = 0 is called a quadratic equation in one variable, where a, b, c are real numbers and a ≠ .
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Solution:
(i) Let the breadth of the rectangular plot = x m
Then, the length of the plot =(2x + 1) m.
Formula of area of rectangle = length × breadth
Area of rectangular plot =528 m² (given)
Putting the value of length and width, we get
⇒(2 x + 1) × x = 528
⇒ 2 x² + x =528
⇒ 2 x² + x – 528 = 0
Since ,it is of the form ax² + bx+c=0, where a≠0.
Thus it represents the required quadratic equation.
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(ii)
Let the first integer = x
Next consecutive positive integer will = x + 1
A.T.Q
⇒x × (x +1) = 306
⇒ x² + x = 306
⇒ x²+ x – 306 = 0
Since ,it is of the form ax² + bx+c=0, where a≠0.
Thus it represents the required quadratic equation.
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(iii)
Let the Rohan’s age = x years
Then, his mother’s age = (x + 26)yr
After 3 years from now
Rohan’s age = (x + 3)yr
Rohan’s mother's age will = (x + 26 + 3 ) = (x + 29)yr
A.T.Q
The product of their ages 3 years from now will be 360 so that
(x + 3)(x + 29) = 360
⇒ x² + 29 x + 3 x + 87 = 360
⇒ x² + 32 x + 87 – 360 = 0
⇒ x² + 32 x – 273 = 0
Since ,it is of the form ax² + bx+c=0, where a≠0.
Thus it represents the required quadratic equation.
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(iv)
1st case:
Let the speed of train be x km/h.
Distance travelled by the train= 480 km
Time taken to travel 480 km = 480/x h ( time= distance/speed)
2nd case:
Let the speed of train = (x – 8) km/h
Distance is same in both the cases
Time taken to travel 480 km = (480/x -8) h
A.T.Q
(480/ X-8 ) =3 + (480 /X)
(480/ X-8 ) - (480 /X) =3
(480X- 480(X-8)) /X(X-8) =3
480 X- 480 X+3840 = 3 X(X-8)
⇒3840 = 3 X² -24 X
⇒3 X² -24 X - 3840 = 0
⇒3 ( X² -8X -1280) =0
Since ,it is of the form ax² + bx+c=0, where a≠0.
Thus it represents the required quadratic equation.
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Hope this will help you.....
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
(2x + 1) × x = 528
⇒ 2x2 + x =528
⇒ 2x2 + x - 528 = 0
(ii)
Let the first integer number = x
Next consecutive positive integer will = x + 1
Product of both integers = x × (x +1) = 306
⇒ x2 + x = 306
⇒ x2 + x - 306 = 0
(iii)
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)
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
⇒ 3x2 - 8x - 1280 = 0