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 :

If the difference between a number and its reciprocal is ²⁴⁄₅, find the number.

Problem 3 :

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 4 :

Divide 25 in two parts so that sum of their reciprocals is ⅙.

Problem 5 :

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 6 : 

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

1. Answer :

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

Subtract x from both sides.

x² - 25x + 144 = 0

x² - 9x - 16x + 144 = 0

x(x - 9) - 16(x - 9) = 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

Therefore, the required number is 16.

2. Answer :

Let y be the required number.

Then, its reciprocal is ¹⁄y.

Given : Difference between a number and its reciprocal is ²⁴⁄₅.

y - ¹⁄y = ²⁴⁄₅

Multiply both sides by 5y to get rid of the denominators y and 5.

5y(y - ¹⁄y) = 5y(²⁴⁄₅)

5y(y) - 5y(¹⁄y) = 24y 

5y² - 5 = 24y

Subtract 24y from boths sides.

5y² - 24y - 5 = 0

Solve by factoring.

5y² - 25y + y - 5 = 0

5y(y - 5) + 1(y - 5) = 0

(y - 5)(5y + 1) = 0

y - 5 = 0  or  5y + 1 = 0

y = 5  or  y = ⁻¹⁄₅

Justification y - 5 = 0  or

Both the values y = 5  and y = ⁻¹⁄₅ satisfy the condition given in the question.

Therefore, the required number is 5 or ⁻¹⁄₅.

3. Answer :

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

Multiply both sides by x(x - 2) to get rid of the denominators (x - 2) an x.

x(x - 2)[⁶⁰⁄₍ₓ ₋ ₂₎ - ⁶⁰⁄ₓ] = x(x - 2)

x(x - 2)[⁶⁰⁄₍ₓ ₋ ₂₎] - x(x - 2)[⁶⁰⁄ₓ] = x(x - 2)

60x - 60(x - 2) = x² - 2x

60x - 60x + 120 = x² - 2x

120 = x² - 2x

0 = x² - 2x - 120 

x² - 2x - 120 = 0

Solve by factoring.

x² - 12x + 10x - 120 = 0

x(x - 12) + 10(x - 12) = 0

(x - 12)(x + 10) = 0

x - 12 = 0  or  x + 10 = 0

x = 12  or  x = -10

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

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

4. Answer :

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. 

¹⁄ₓ + ¹⁄₍₂₅ ₋ ₓ₎ = ⅙

Multiply both sides by 6x(25 - x) to get rid of the denominators x, (25 - x) and 6.

6x(25 - x)[¹⁄ₓ + ¹⁄₍₂₅ ₋ ₓ₎] = 6x(25 - x)[⅙]

6x(25 - x)[¹⁄ₓ] - 6x(25 - x)[¹⁄₍₂₅ ₋ ₓ₎] = x(25 - x)

6(25 - x) - 6x = 25x - x²

150 - 6x + 6x = 25x - x²

150 = 25x - x²

x² - 25x + 150 = 0

Solve by factoring.

x² - 15x - 10x + 150 = 0

x(x - 15) - 10(x - 15) = 0

(x - 15)(x - 10) = 0

x - 15 = 0  or x - 10 = 0

x = 15  or  x = 10

When x = 15, 

25 - x = 25 - 15

= 10

When x = 10,

25 - x = 25 - 10

= 15

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

5. Answer :

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

Solve by factoring.

x² - 12x + 16x - 192 = 0

x(x - 12) + 16(x - 12) = 0

(x - 12)(x + 16) = 0

x - 12 = 0  or  x + 16 = 0

x - 12 = 0  or  x + 16 = 0

x = 12  or  x = -16

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

If x = 12,

x + 4 = 12 + 4

= 16

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

6. Answer :

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 - 24x + 144 = 2x² - 54x + 365

0 = x² - 34x + 221

x² - 34x + 221 = 0

Solve by factoring.

x² - 13x - 17x + 221 = 0

x(x - 13) - 17(x - 13) = 0

(x - 13)(x - 17) = 0

x - 13 = 0  or  x - 17 = 0

x = 13  or  x = 17 

x = 13 can not be accepted.

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

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

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