Question

Represent the following situations in the form of quadratic equation: (a) The area of a rectangular plot is 465 m^2. The length of the plot is one more than twice its breadth. We find find to the length and breadth of the plot? ​

Answer 1

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

Product of two consecutive integers = 306 (given)

⇒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

⇒X² -8 X -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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(i)

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)

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