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Math Homework Help | Math Homework Help Service
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Copyright 息 2014-2016 Homework1.com, All rights reserved
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Sample of Math Homework Help Illustrations and Solutions:
Example: If the root of the equation 22
- 10x +  = 0 = is 2, then the value of  is :
(a) -3 (b) -6, (c) 9 (d) 12
Solution. Since is a root of the equation
 2(2)2
- 10(2) +  =0 =  = 20  8 = 12  (d) holds.
Example: Which of the following is a root of the equation 22
- 5x  3 = 0 ?
(a) x = 3 (b) x = 4 (c) x = 1 (d) x = -4
Solution. (a) holds. ( 2(3)2
-5(3)  3 = 0 = 18  15  3 = 0)
Example: If no root of 2
-kx + 1 = 0 is real, then
(a) -3 < k < 3 (b) 2 <k <2 (c) k > 2 (d) k < -2
(b) Solution. Since the roots of 2
 kx + 4 = 0 are non-real.
 Disc. ()2
- 4 < 0 = 2
- 4 < 0 = 2
< 4 = |k| <2  (b) holds
Example: If 2
+ 4x + k = 0 has real roots, then
(a) -3 < k < 3 (b) -2 <k <2 (c) k > 2 (d) k < -2
(C.B.S.E. 2012)
Solution. Since the roots of 2
- kx + 4 = 0 are non-real.
 Disc. ()2
-4 < 0 = 2
< 4 = |k| < 2 = -2 < k < 2  (b) holds.
Example: If 2
+ 4x + k = 0 has real roots, then
Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
(a) k  4 (b) k  4 (c) k  0 (d) k  0
Solution. Since 2
+ 4x + k = has real roots.
 Disc. (4)2
 4k  0 = 4k  16 = k  4  (b) holds.
Example: Value of k for which the quadratic equation 22
- kx + k = 0 has equal roots is
(a) 0 only (b)4 (c) 8 only (d) 0.8.
(N.C.E.R.T. Exemplar Problem)
Solution. For equal roots, Disc. = ()2
-4(2) (k) = 0 = k (k  8) = 0 = k = 0.8
 (d) holds.
Example: The value of k for which 32
+ 2x + k = 0 has real roots is :
(a) k >
1
3
(b) k 
1
3
(c) k 
1
3
(d) k <
1
3
Solution. For real roots, Disc. = (2)2
 4(3) (k)  0
= 4  12k  0 = 4  12 k = 12 k  4 = k 
4
12
=
1
13
=  (b) holds.
Example: If the quadratic equation 2
+ 2x + m = 0 has two equal roots, then the
values of m are
(a) 賊 1 (b) 0,-2 (c) 0,1 (d) -1,0
Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Solution. Disc. = 2
- 4 (6) (2) = 1 (given) = 2
= 48 + 1 = 49 =b = 賊 7  (c) holds.
Solved examples  6
Example: Find the sum and product of roots of (-2)2
+ 5x + 4 = 0.
Solution. Sum of roots =


=
5
2
=
5
2
; product of roots =


=
4
2
=-2
Example: If  and  are roots of 92
- 24x + 8 = 0, then find  +  and 腫.
Solution.  +  =


=
(24)
9
=
8
3
; 腫 =


=
8
9
Example: Form a quadratic equation roots -
1
3
and
5
2
.
Solution. S = Sum of roots = 
1
3
+
5
2
=
2+15
6
; P = Product of roots = 
1
3
5
2
= -
5
6
 equation is 2
 Sx + p = 0 = 2
-
13
6
x -
5
6
= 0 = 62
-13x  5 = 0
Example: If one root of quadratic equation with rational co-efficients is 2 + 3 , then
give other root.
Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Solution. Note that in a quadratic equation with rational co-efficients, surd roots occur
in conjugate pairs. = 2 - 3 is the other root.
1. The sum of the squares of two positive integers is 208. If the square of the larger
number is 18 times the smaller, find the numbers.
Solution. Let the smaller number = x ; Square of larger number = 18x
Given, 2
+ 18x = 208 = 2
+ 18x  208 = 0
= (x  8) (x + 26) = 0 =x = 8, -26 = x = 8
 Square of larger number = 18x = (18) (8) = 144
= Larger number = 144 = 12 and smaller number = x = 8
2. If -5 is root of quadratic equation 22
+ 2px  15 = 0 and the quadratic equation
 (2
+ x) k = 0 has equal roots, find the value of k.
Solution. 5 is root of 22
+ 2px  15 = 0 = 2 (5)2
+ 2 (-5)  15 = 0 =  =
7
2
  (2
+ x) + k = 0 =
7
2
(2
+ x) + k = 0 = 72
+ 7x + 2k = 0
It has equal roots = D = 0 = (7)2
-4(7) (2k) = 0 = k =
7
8
3. If, p, q, r and s are real numbers such that pr = 2(q + s), then show that at least
one of the equations 2
+ px + q = 0 has real roots.
4. If the roots of the equation 2
+ 2cx + ab = 0 are real and unequal, prove that
the equation 2
-2 (a + b) x + 2
+ 2
+ 22
= 0 has no real roots.
5. A person on tour has $360 for his expenses. If the extends his tour for 4 days, he
has to cut down his daily expenses by $3. Find the original duration of the tour.
Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Solution. Let the original duration of the tour be x days.
 total expenditure on tour = $360  expenditure per day = $
360

Duration of the extended tour = (x + 4) days
 expenditure per day according to new schedule = $
360
+4
Since the daily expenses are cut down by $3.

360

-
360
+4
= 3 =
360 +4  360 
(+4)
= 3
=
360 +1440360 
 (+4)
= 3 =
1440
2+ 4
= 3 = 2
+ 4x = 480 = 2
+ 4x  480 = 0
= 2
+ 24x  20x  480 = 0 = x (x + 24) -20 (x + 24) = 0
= (x  20) (x + 24) = 0 = x  20 = 0 or, x + 24 = 0 =x = 20 or, x = -24
But, the number of days cannot be negative. So, x = 20
Hence, the original duration of the tour was of 20 days.
6. (a) A shopkeeper buys a number of books for $80. If he had bought 4 more
books for the same amount, each book would have cost $12 less. How many
books did he buy?
Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
(b) A shopkeeper buys a number of books for $1200. If he had bought 10 more books
for the same amount, each book would have cost him $20 less. How many books did he
buy ?
Solution. (a) Let number of books bought be x. Then, Cost of x books = $80
= Cost of one book = $
80

If the number of books bought is x + 4, then Cost of one book = $
80
+4

It is given that the cost of one book is reduced by one rupee

80

-
80
+4
= 1 = 80
1


1
+4
= 1 = 80
+4
 +4
= 1 =
320
2+ 4
= 1
= 2
+ 4x = 320 = 0 = 2
+ 20x  16x  320 = 0 = x (x + 20)  16(x + 20) = 0
= (x + 20) (x  16) = 0 =x = -20 or, x = 16 = x = 16 [ x cannot be negative]
Hence, the number of books is 16.
(b) Similar to Part (a) Ans. 20.
7. If the price of a book is reduced by $5, a person can buy 5 more books for $300. Find
the original list price of the book.
Solution. Let the original list price of the book be $ x  number of books bought for
$300 =
300

Reduced list price of the book = $(x -5)  number of books bought for $300 =
300
モ5
By the given condition,
300
モ5
-
300

= 5 =
300 モ300 +1500
(モ5)
= 5 =
1500
25
= 5
= 2
- 5x = 300 = 2
- 5x  300 = 0 = 2
- 20x + 15x  300 = 0 = (x  20) (x + 15) =
0
= x  20 = 0 or, x + 15 = 0 = x = 20, x = -15 =x = 20 [ x = -15 is not possible]
Hence, the list price of the books = $20
8. A factory kept increasing is doubled in the last two year. Find the percentage if it is
known that the output is doubled in the last two years.
Solution. Let p be the initial production (2 years ago), and let the increase in product
every year be x%. Then,
Product at the end of first year = P +

100
= p 1 +

100
Product at the end of the second year
= P 1 +

100
+

100
1 +

100
= P 1 +

100
1 +

100
= P 1 +

100
2
Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Since product is doubled in last two years  P 1 +

100
2
= 2P = 1 +

100
2
= 2
= (100 + )2
= 2  1002
= 2
+ 200x  10000 = 0
= x =
200 賊 (200)2+ 40000
2
= - 100 賊 100 2 = 100 (-1 + 2)
= x = 100 (-1 + 2) [ x cannot be negative]
9. The difference of the ages of Sohrab and his father is 30 years. If the difference of the
squares of their ages is 1560, find their ages.
Solution. Let y, x be the ages of Soharb and his father respectively
 x  y = 30 (1)
and 2
- 2
= 1560 (2)
Divide (2) by (1), we get
2 2
モ
=
1560
30
= x + y = 52 (3)
(1) + (2) gives 2x = 82 =x = 41. (3)  (1) gives 2y = 22 = y =
11
Hence required ages are 41 years, 11 years.

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Help with math homework

  • 1. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved Math Homework Help | Math Homework Help Service Contact Us Homework1 3422 SW 15 Street Suite #8924 Deerfield Beach, FL, US 33442 Tel: +1-626-472-1732 Web: https://homework1.com/ Email: info@homework1.com Facebook: https://www.facebook.com/homework1com Linkedin: https://www.linkedin.com/in/homework1 Twitter: https://twitter.com/homework1_com Google Plus: https://plus.google.com/+Homework1/ Pinterest: https://www.pinterest.com/homeworkone/
  • 2. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved About Us: At Homework1.com we offer authentic and 100% accurate online homework help and study assistance to students from USA, UK, Australia, and Canada. However, we dont offer students only academic assignment help service to complete their study project; rather we offer our best effort to teach our student-clients about the assignment we have solved. Our tutors are not only subject matter experts, they are avid student-mentors and are ready to walk extra miles to make them understand the fundamentals of the assignment done, and help them to learn the solution by heart. We are always ready to hear from you as you will find us always a few clicks away in 247! We are reachable by a phone calls you can send us an email or simply but accessing our site you can call us for a live chat! We are ready to help you at the most critical hour of your assignment submission and will take care of your task with best sincerity and prompt turnaround time! Help with math homework service proves to be beneficial at exam times and assignments having short deadlines. Sample of Math Homework Help Illustrations and Solutions: Example: If the root of the equation 22 - 10x + = 0 = is 2, then the value of is : (a) -3 (b) -6, (c) 9 (d) 12 Solution. Since is a root of the equation 2(2)2 - 10(2) + =0 = = 20 8 = 12 (d) holds. Example: Which of the following is a root of the equation 22 - 5x 3 = 0 ? (a) x = 3 (b) x = 4 (c) x = 1 (d) x = -4 Solution. (a) holds. ( 2(3)2 -5(3) 3 = 0 = 18 15 3 = 0) Example: If no root of 2 -kx + 1 = 0 is real, then (a) -3 < k < 3 (b) 2 <k <2 (c) k > 2 (d) k < -2 (b) Solution. Since the roots of 2 kx + 4 = 0 are non-real. Disc. ()2 - 4 < 0 = 2 - 4 < 0 = 2 < 4 = |k| <2 (b) holds Example: If 2 + 4x + k = 0 has real roots, then (a) -3 < k < 3 (b) -2 <k <2 (c) k > 2 (d) k < -2 (C.B.S.E. 2012) Solution. Since the roots of 2 - kx + 4 = 0 are non-real. Disc. ()2 -4 < 0 = 2 < 4 = |k| < 2 = -2 < k < 2 (b) holds. Example: If 2 + 4x + k = 0 has real roots, then
  • 3. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved (a) k 4 (b) k 4 (c) k 0 (d) k 0 Solution. Since 2 + 4x + k = has real roots. Disc. (4)2 4k 0 = 4k 16 = k 4 (b) holds. Example: Value of k for which the quadratic equation 22 - kx + k = 0 has equal roots is (a) 0 only (b)4 (c) 8 only (d) 0.8. (N.C.E.R.T. Exemplar Problem) Solution. For equal roots, Disc. = ()2 -4(2) (k) = 0 = k (k 8) = 0 = k = 0.8 (d) holds. Example: The value of k for which 32 + 2x + k = 0 has real roots is : (a) k > 1 3 (b) k 1 3 (c) k 1 3 (d) k < 1 3 Solution. For real roots, Disc. = (2)2 4(3) (k) 0 = 4 12k 0 = 4 12 k = 12 k 4 = k 4 12 = 1 13 = (b) holds. Example: If the quadratic equation 2 + 2x + m = 0 has two equal roots, then the values of m are (a) 賊 1 (b) 0,-2 (c) 0,1 (d) -1,0
  • 4. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved Solution. Disc. = 2 - 4 (6) (2) = 1 (given) = 2 = 48 + 1 = 49 =b = 賊 7 (c) holds. Solved examples 6 Example: Find the sum and product of roots of (-2)2 + 5x + 4 = 0. Solution. Sum of roots = = 5 2 = 5 2 ; product of roots = = 4 2 =-2 Example: If and are roots of 92 - 24x + 8 = 0, then find + and 腫. Solution. + = = (24) 9 = 8 3 ; 腫 = = 8 9 Example: Form a quadratic equation roots - 1 3 and 5 2 . Solution. S = Sum of roots = 1 3 + 5 2 = 2+15 6 ; P = Product of roots = 1 3 5 2 = - 5 6 equation is 2 Sx + p = 0 = 2 - 13 6 x - 5 6 = 0 = 62 -13x 5 = 0 Example: If one root of quadratic equation with rational co-efficients is 2 + 3 , then give other root.
  • 5. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved Solution. Note that in a quadratic equation with rational co-efficients, surd roots occur in conjugate pairs. = 2 - 3 is the other root. 1. The sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller, find the numbers. Solution. Let the smaller number = x ; Square of larger number = 18x Given, 2 + 18x = 208 = 2 + 18x 208 = 0 = (x 8) (x + 26) = 0 =x = 8, -26 = x = 8 Square of larger number = 18x = (18) (8) = 144 = Larger number = 144 = 12 and smaller number = x = 8 2. If -5 is root of quadratic equation 22 + 2px 15 = 0 and the quadratic equation (2 + x) k = 0 has equal roots, find the value of k. Solution. 5 is root of 22 + 2px 15 = 0 = 2 (5)2 + 2 (-5) 15 = 0 = = 7 2 (2 + x) + k = 0 = 7 2 (2 + x) + k = 0 = 72 + 7x + 2k = 0 It has equal roots = D = 0 = (7)2 -4(7) (2k) = 0 = k = 7 8 3. If, p, q, r and s are real numbers such that pr = 2(q + s), then show that at least one of the equations 2 + px + q = 0 has real roots. 4. If the roots of the equation 2 + 2cx + ab = 0 are real and unequal, prove that the equation 2 -2 (a + b) x + 2 + 2 + 22 = 0 has no real roots. 5. A person on tour has $360 for his expenses. If the extends his tour for 4 days, he has to cut down his daily expenses by $3. Find the original duration of the tour.
  • 6. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved Solution. Let the original duration of the tour be x days. total expenditure on tour = $360 expenditure per day = $ 360 Duration of the extended tour = (x + 4) days expenditure per day according to new schedule = $ 360 +4 Since the daily expenses are cut down by $3. 360 - 360 +4 = 3 = 360 +4 360 (+4) = 3 = 360 +1440360 (+4) = 3 = 1440 2+ 4 = 3 = 2 + 4x = 480 = 2 + 4x 480 = 0 = 2 + 24x 20x 480 = 0 = x (x + 24) -20 (x + 24) = 0 = (x 20) (x + 24) = 0 = x 20 = 0 or, x + 24 = 0 =x = 20 or, x = -24 But, the number of days cannot be negative. So, x = 20 Hence, the original duration of the tour was of 20 days. 6. (a) A shopkeeper buys a number of books for $80. If he had bought 4 more books for the same amount, each book would have cost $12 less. How many books did he buy?
  • 7. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved (b) A shopkeeper buys a number of books for $1200. If he had bought 10 more books for the same amount, each book would have cost him $20 less. How many books did he buy ? Solution. (a) Let number of books bought be x. Then, Cost of x books = $80 = Cost of one book = $ 80 If the number of books bought is x + 4, then Cost of one book = $ 80 +4 It is given that the cost of one book is reduced by one rupee 80 - 80 +4 = 1 = 80 1 1 +4 = 1 = 80 +4 +4 = 1 = 320 2+ 4 = 1 = 2 + 4x = 320 = 0 = 2 + 20x 16x 320 = 0 = x (x + 20) 16(x + 20) = 0 = (x + 20) (x 16) = 0 =x = -20 or, x = 16 = x = 16 [ x cannot be negative] Hence, the number of books is 16. (b) Similar to Part (a) Ans. 20. 7. If the price of a book is reduced by $5, a person can buy 5 more books for $300. Find the original list price of the book. Solution. Let the original list price of the book be $ x number of books bought for $300 = 300 Reduced list price of the book = $(x -5) number of books bought for $300 = 300 モ5 By the given condition, 300 モ5 - 300 = 5 = 300 モ300 +1500 (モ5) = 5 = 1500 25 = 5 = 2 - 5x = 300 = 2 - 5x 300 = 0 = 2 - 20x + 15x 300 = 0 = (x 20) (x + 15) = 0 = x 20 = 0 or, x + 15 = 0 = x = 20, x = -15 =x = 20 [ x = -15 is not possible] Hence, the list price of the books = $20 8. A factory kept increasing is doubled in the last two year. Find the percentage if it is known that the output is doubled in the last two years. Solution. Let p be the initial production (2 years ago), and let the increase in product every year be x%. Then, Product at the end of first year = P + 100 = p 1 + 100 Product at the end of the second year = P 1 + 100 + 100 1 + 100 = P 1 + 100 1 + 100 = P 1 + 100 2
  • 8. Homework1 Copyright 息 2014-2016 Homework1.com, All rights reserved Since product is doubled in last two years P 1 + 100 2 = 2P = 1 + 100 2 = 2 = (100 + )2 = 2 1002 = 2 + 200x 10000 = 0 = x = 200 賊 (200)2+ 40000 2 = - 100 賊 100 2 = 100 (-1 + 2) = x = 100 (-1 + 2) [ x cannot be negative] 9. The difference of the ages of Sohrab and his father is 30 years. If the difference of the squares of their ages is 1560, find their ages. Solution. Let y, x be the ages of Soharb and his father respectively x y = 30 (1) and 2 - 2 = 1560 (2) Divide (2) by (1), we get 2 2 モ = 1560 30 = x + y = 52 (3) (1) + (2) gives 2x = 82 =x = 41. (3) (1) gives 2y = 22 = y = 11 Hence required ages are 41 years, 11 years.