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Question 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

Solutions:

(i) Given,

(x + 1)2 = 2(x – 3)

By using the formula for (a+b)=a2+2ab+b2

x2 + 2x + 1 = 2x – 6

x2 + 7 = 0

Since the above equation is in the form of ax2 + bx + c =0.

Therefore, the given equationis quadratic equation.

(ii) Given, x2 – 2x = (–2) (3– x)

By using the formula for (a+b)=a2+2ab+b2

x 2x = -6 + 2x

x– 4x + 6 = 0

Since the above equation is in the form of ax2 + bx + c =0.

Therefore, the given equationis quadratic equation.

(iii) Given, (x – 2)(x + 1) = (x – 1)(x + 3)

By using the formula for (a+b)=a2+2ab+b2

x– x – 2 = x+2x – 3

3x – 1 = 0

Since the above equation is not in the form ofax2 + bx + c = 0.

Therefore, the given equation is not aquadratic equation.

(iv) Given, (x – 3)(2x +1) = x(x + 5)

By using the formula for (a+b)2=a2+2ab+b2

2x– 5x – 3 = x+5x

x– 10x – 3 = 0

Since the above equation is in the form of ax2 + bx + c =0.

Therefore, the given equationis quadratic equation.

(v) Given, (2x – 1)(x – 3) =(x + 5)(x – 1)

By using the formula for (a+b)2=a2+2ab+b2

2x– 7x + 3 = x+4x – 5

x– 11x + 8 = 0

Since the above equation is in the form of ax2 + bx + c =0.

Therefore, the given equationis quadratic equation.

(vi) Given, x2 + 3x + 1 =(x – 2)2

By using the formula for (a+b)2=a2+2ab+b2

x2 + 3x + 1 = x2 +4 – 4x

7x – 3 = 0

Since the above equation is not in the form ofax2 + bx + c = 0.

Therefore, the given equation is not aquadratic equation.

(vii) Given, (x + 2)3 =2x(x2 – 1)

By using the formula for (a+b)=a2+2ab+b2

x3 + 8 + x2 +12x = 2x3 – 2x

x3 + 14x – 6x2 – 8 =0

Since the above equation is not in the form ofax2 + bx + c = 0.

Therefore, the given equation is not aquadratic equation.

(viii) Given, x3 – 4x2 – x +1 = (x – 2)3

By using the formula for (a+b)=a2+2ab+b2

x3 – 4x2 – x +1 = x3 – 8 – 6x + 12x

2x2 – 13x + 9 = 0

Since the above equation is in the form of ax2 + bx + c =0.

Question 2 : Represent the following situations in the form of quadratic equations:

(i) The area of arectangular plot is 528 m2. The length of the plot (in metres) isone more than twice its breadth. We need to find the length and breadth of theplot.

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

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

(iv) A train travelsa distance of 480 km at a uniform speed. If the speed had been 8 km/h less, thenit would have taken

Solutions:

(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.

Solution

(i) Let us consider,

Breadth of the rectangular plot= x m

Thus, the length of the plot = (2x + 1)m.

As we know,

Area of rectangle = length × breadth= 528 m2

Putting the value of length and breadth of theplot in the formula, we get,

(2x + 1) × x = 528

2x2 + x =528

2x2 + x – 528 = 0

Therefore, the length and breadth of plot,satisfies the quadratic equation, 2x2 + x – 528 = 0,which is the required representation of the problem mathematically.

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

Solution

(ii) Let us consider,

The first integer number = x

Thus, the next consecutive positive integerwill be = x + 1

Product of two consecutive integers= x × (x +1) = 306

x+ x = 306

x+ x – 306 = 0

Therefore, the two integers x and x+1,satisfies the quadratic equation, x+ x – 306 = 0,which is the required representation of the problem mathematically.

(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

(iii) Let us consider,

Age of Rohan’s = x  years

Therefore, as per the given question,

Rohan’s mother’s age = x + 26

After 3 years,

Age of Rohan’s = x + 3

Age of Rohan’s mother will be = x +26 + 3 = x + 29

The product of their ages after 3 years willbe equal to 360, such that

(x + 3)(x + 29) = 360

x2 + 29x + 3x + 87 = 360

x2 + 32x + 87 – 360 = 0

x2 + 32x – 273 = 0

Therefore, the age of Rohan and his mother,satisfies the quadratic equation, x2 + 32x – 273 = 0,which is the required representation of the problem mathematically.

(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

Solution

(iv) Let us consider,

The speed of train = x  km/h

And

Time taken to travel 480 km = 480/x km/hr

As per second condition, the speed of train =(x – 8) km/h

Also given, the train will take 3 hours tocover the same distance.

Therefore, time taken to travel 480 km =480/(x+3) km/h

As we know,

Speed × Time = Distance

Therefore,

(x – 8)(480/(x + 3) =480

480 + 3x – 3840/x – 24 = 480

3x – 3840/x = 24

3x– 8x – 1280 = 0

Therefore, the speed of the train, satisfiesthe quadratic equation, 3x– 8x – 1280 = 0,which is the required representation of the problem mathematically.

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