QUADRATIC EQUATION WORD PROBLEMS WORKSHEET WITH ANSWERS

Problem 1 :

Difference between a number and its positive square root is 12. Find the number. 

Problem 2 :

A piece of iron rod costs $60. If the rod was 2 meter shorter and each meter costs $1 more, the cost would remain unchanged. What is the length of the rod ? 

Problem 3 :

Divide 25 in two parts so that sum of their reciprocals is 1/6.

Problem 4 :

The hypotenuse of a right angled triangle is 20 cm. The difference between its other two sides is 4 cm. Find the length of the sides.  

Problem 5 : 

The sides of an equilateral triangle are shortened by 12 units, 13 units and 14 units respectively and a right angle triangle is formed. Find the length of each side of the equilateral triangle.   

Answers

Problem 1 :

Difference between a number and its positive square root is 12. Find the number. 

Solution :

Let "x" be the required number.

Its positive square root is √x 

Given : Difference between x and √x is 12.

x - √x  =  12

x - 12  =  √x

(x - 12)2  =  x

x2 - 24x + 144  =  x

x2 - 25x + 144  =  0

(x - 9)(x - 16)  =  0

x  =  9  or  x  =  16

x  =  9 does not satisfy the condition given in the question.

Then, 

x  =  16

So, the required number is 16.

Problem 2 :

A piece of iron rod costs $60. If the rod was 2 meter shorter and each meter costs $1 more, the cost would remain unchanged. What is the length of the rod ? 

Solution :

Let "x" be the length of the given rod. 

Then the length of the rod 2 meter shorter is (x - 2) and the total cost of both the rods is $60 (Because cost would remain unchanged). 

Cost of one meter of the given rod  is

=  60 / x 

Cost of one meter of the rod which is 2 meter shorter is

=  60 / (x - 2) 

Given : If the rod was 2 meter shorter and each meter costs $1 more.

That is, 60/(x-2) is $1 more than 60/x.

 [60 / (x - 2)]  -  [60 / x]  =  1 

Simplify.

[60x - 60(x - 2)]  /  [x(x - 2)]  =  1

[60x - 60x + 120]  /  [x² - 2x]  =  1

120  /  (x2 - 2x)  =  1

120  =  x2 - 2x

0  =  x2 + 2x - 120 

x2 + 2x - 120  =  0

(x + 10)(x - 12)  =  0 

x  =  - 10  or  x  =  12

Because length can not be a negative number, we can ignore "- 10". 

So, the length of the given rod is 12 m.

Problem 3 :

Divide 25 in two parts so that sum of their reciprocals is 1/6.

Solution :

Let "x" be one of the parts of 25. Then the other part is (25 - x). 

Given : Sum of the reciprocals of the parts is 1/6. 

Then, we have

1/x  +  1/(25 - x)  =  1/6

Simplify.

(25 - x + x) / x(25 - x)  =  1/6 

25 / (25x - x2)  =  1/6 

6(25)  =  25x - x2 

150  =  25x - x2 

x2 - 25x + 150  =  0 

(x - 15)(x - 10)  =  0 

x  =  15  or  x  =  10 

When x  =  15, 

25 - x  =  25 - 15

25 - x  =  10

When x  =  10, 

25 - x  =  25 - 10

25 - x  =  15

So, the two parts of the 25 are 10 and 15.

Problem 4 :

The hypotenuse of a right angled triangle is 20 cm. The difference between its other two sides is 4 cm. Find the length of the sides.  

Solution :

Let "x" and "x + 4" be the lengths of other two sides. 

Using Pythagorean theorem, we have

(x + 4)2 + x2  =  202

Simplify.

x2 + 8x + 16 + x2  =  400 

2x2 + 8x + 16  =  400

Subtract 400 from both sides. 

2x2 + 8x - 384  =  0

Divide both sides by 2. 

x2 + 4x - 192  =  0 

(x + 16)(x - 12)  =  0 

x  =  -16 or x  =  12 

x = -16 can not be accepted. Because length can not be negative. 

If x  =  12,  

x + 4  =  12 + 4  =  16 

So, the other two sides of the triangle are 12 cm and 16 cm.

Problem 5 : 

The sides of an equilateral triangle are shortened by 12 units, 13 units and 14 units respectively and a right angle triangle is formed. Find the length of each side of the equilateral triangle.   

Solution :

Let "x" be the length of each side of the equilateral triangle. 

Then, the sides of the right angle triangle are

(x - 12), (x - 13) and (x - 14) 

In the above three sides, the side represented by (x - 12) is hypotenuse (Because that is the longest side). 

Using Pythagorean theorem, we have

(x - 12)2  =  (x - 13)2 + (x - 14)2

 x2 - 24x + 144  =  x2 - 26x + 169 + x2 - 28x + 196 

x2 - 30x + 221  =  0 

(x - 13)(x - 17)  =  0 

x  =  13  or  x  =  17. 

x  =  13 can not be accepted.

Because, if x  =  13, the side represented by (x - 14) will be negative. 

So, the length of each side of the equilateral triangle is 17 units.

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