SOLVING WORD PROBLEMS USING QUADRATIC EQUATIONS

Problem 1 :

The area of a rectangular plot is 528 m². The length of the plot (in meters) is one more than twice its width. Find the length and width of the plot.

Solution :

Let x be the width of the plot.

Then the length of the plot is

= 2x + 1

Area of rectangular plot = 528 m².

length ⋅ width = 528

(2x + 1)x = 528

2x² + x = 528

2x² + x - 528 = 0

Solving the above quadratic equation, we get

x = 16  or  x = -16.5

Since x represents width of the rectangle, it can not have a negative value. So, we can ignore x = -16.5.

Therefore, x = 16.

Length = 16 m

Width = 2(16) + 1

= 32 + 1

= 33 m

Problem 2 :

If the product of two consecutive positive integers is 306. then find the integers.

Solution :

Let x and x + 1 be the two consecutive positive integers.

Product of these integers = 306

 x(x + 1) = 306

x² + x = 306

 x² + x - 306 = 0

Solving the above quadratic equation, we get

x = 17  or  x = -18

Since x is a positive integer, we can ignore x = -18.

x = 18

x + 1 = 18 + 1

= 19

Therefore, the two consecutive positive integers are 18 and 19.

Problem 3 :

Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. Find the present age of Rohan.

Solution :

Let x be Rohan's present age.

Then, the present age of Rohan's mother is

= x + 26

Three years later,

Rohan's age = x + 3

Mother's age = x + 26 + 3

Product of their ages = 360

(x + 3)(x + 29) = 360

x² + 29x + 3x + 87 = 360

x² + 32x + 87 = 360

x² + 32x + 87 - 360 = 0

x² + 32x - 273 = 0

Solving the above quadratic equation, we get

x = 7  or  x = -39

Since x represents the present age of Rohan, we can ignore x = -39.

Therefore, the present age of Rohan is 7 years.

Problem 4 :

A train travels 480 km of distance at a uniform speed. If the speed is reduced by 8 km/hr, then it takes 3 hours more to cover the same distance. Find the speed of the train.

Solution :

Let x be the original speed of the train.

Time = Distance/Speed

Time taken by the train to cover 480 km distance at the speed of x km/hr :

= 480/x

Time taken by the train to cover 480 km distance in the reduced speed, that is (x - 8) km/hr :

= 480/(x - 8)

Given : When the speed is (x - 8) km/hr, it takes 3 hours more to cover the same distance.

480/(x - 8) = (480/x) + 3

480/(x - 8) = (480 + 3x)/x

480x = (3x + 480)(x - 8)

480x = 3x² - 24x + 480x - 3840

Subtract 480x from both sides.

0 = 3x² - 24x - 3840

Divide both sides by 3.

0 = x² - 8x - 1280

Solving the above quadratic equation, we get

x = 40  or  x = -32

Since x represents the original speed of the train, it can not be negative. So, we can ignore x = -32.

Therefore, original speed of the train is 40 km/hr.

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