[quadratic word problems part 2][section 68]

(1.) The square of a number is 56 more than the number itself.

What is the number?

n² = n + 56 here is the problem

-n -n subtract n from each side
_______________
n² - n = 56 subtract

-56 -56 subtract 56 from each side
_______________
n² - n - 56 = 0 subtract

(n - 8)(n + 7) = 0 factor

n - 8 = 0 n + 7 = 0 set each factor equal to 0

+8 +8 - 7 -7 add this to each side
____________ __________
n = 8 , n = -7 add

results: n = 8 , n = -7

(2.) The perimeter of a rectangle is 30 meter and its area

is 54 square meters. What are its dimensions?

2L + 2w = 30

wL = 54 here is the problem

2L + 2w = 30
___ __ ___
2 2 2 divide thru by 2

L + w = 15 divide and cancel

-w -w subtract w from each side
____________________
L = 15 - w subtract

w(15 - w) = 54 replace L with 15 - w

15w - w² = 54 multiply thru parentheses

-15w + w² = -54 multiply thru by -1

w² - 15w = -54 rearrange terms

+ 54 +54 add 54 to each side
____________________
w² - 15w + 54 = 0 add

(w - 9)(w - 6) = 0 factor

w - 6 = 0 set this factor equal to 0

+ 6 +6 add 6 to each side
______________
w = 6 add

L = 15 - 6 replace w with 6

L = 9 subtract

results: w = 6 and L = 9

(3.) A number exceeds its square by 2/9. Find the number.

n = n² + (2/9) here is the problem

9n = 9n² + 2 multiply thru by 9, cancel

9n² + 2 = 9n just rearrange like this

-9n -9n subtract 9n from each side
________________
9n² - 9n + 2 = 0 subtract

(3n - 1)(3n - 2) = 0 factor

3n - 1 = 0 3n - 2 = 0 set each factor equal to 0

+ 1 +1 + 2 +2 add this to each side
_______________ ____________
3n = 1 , 3n = 2 add
___ ___ ___ ___
3 3 3 3 divide each side by 3

n = 1/3 , n = 2/3 cancel

results: n = 1/3, n = 2/3

(4.) The difference of the squares of two consecutive odd

integers is 56. What are the integers?

n² - (n + 2)² = 56 here is the problem

n² - (n² + 4n + 4) = 56 square the binomial

n² - n² - 4n - 4 = 56 multiply thru parentheses

-4n - 4 = 56 combine like terms

4n + 4 = -56 multiply thru by -1

-4 -4 subtract 4 from each side
___________________
4n = -60 subtract
___ ____
4 4 divide each side by 4

n = -15 divide and cancel

result:

The numbers are 13 and 15. And the numbers are -15 and -13 .

(5.) A rectangular city lot has an area of 1000 square meters.

If the length of the lot is 10 meters more than twice its

width, find the dimensions of the lot.

wL = 1000

L = 10 + 2w here is the problem

w(10 + 2w) = 1000 replace L with 10 + 2w

10w + 2w² = 1000 multiply thru parentheses

2w² + 10w = 1000 rearrange terms
____ ____ _____
2 2 2 divide thru by 2

w² + 5w = 500 divide and cancel

-500 -500 subtract 500 from each side
____________________
w² + 5w - 500 = 0 subtract

(w + 25)(w - 20) = 0 factor

w - 20 = 0 set this factor equal to 0

+ 20 +20 add 20 to each side
_______________
w = 20 add

L = 10 + 2(20) replace w with 20

L = 50 multiply add

results: w = 20 and L = 50

(6.) The numerator of a fraction is 1 less than its

denominator. If the fraction is increased by 2 times

its reciprocal, the sum will be 3 5/12. Find the

numerator and the denominator.

n = d - 1

(n/d) + 2(d/n) = 3 5/12 here is the problem

(n/d) + (2d/n) = 41/12 write as an improper fraction

12n² + 24d² = 41dn multiply thru by 12dn and cancel

12(d - 1)² + 24d² = 41d(d - 1) replace n with d - 1

12d² - 24d + 12 + 24d² = 41d² - 41d multiply

36d² - 24d + 12 = 41d² - 41d combine like terms

+41d + 41d add 41d to each side
________________________________
36d² + 17d + 12 = 41d² add

-36d² - 17d - 12 = -41d² multiply thru by -1

+ 41d² + 41d² add this to each side
____________________________
5d² - 17d - 12 = 0 add

(5d + 3)(d - 4) = 0 factor

d - 4 = 0 set this factor equal to 0

+ 4 +4 add 4 to each side
________________
d = 4 add

n = 4 - 1 replace d with 4

n = 3 subtract

results: the numerator is 3 and the denominator is 4.

(7.) One number is 3 more than another. The product of

the numbers is 54. Find the numbers.

a = b + 3

ab = 54 here is the problem

(b + 3)(b) = 54 replace a with b + 3

b² + 3b = 54 multiply thru parentheses

-54 -54 subtract 54 from each side
__________________
b² + 3b - 54 = 0 subtract

(b + 9)(b - 6) = 0 factor

b + 9 = 0 b - 6 = 0 set each factor equal to 0

- 9 -9 + 6 + 6 add this to each side
______________ ______________
b = -9 , b = 6 add

a = -9 + 3 a = 6 + 3 replace b with -9 and with 6

a = -6 a = 9 add

results: The numbers are 6 and 9, or, the numbers are -9 and -6.

(8.) The denominator of a fraction is 2 more than its

numerator. If the fraction is increased by 3 times

its reciprocal, the sum will be 5 3/5. Find the

fraction.

d = 2 + n

(n/d) + 3(d/n) = 5 3/5 here is the problem

(n/d) + (3d/n) = (28/5) write as an improper fraction

5n² + 15d² = 28dn multiply thru by 5nd, and cancel

5n² + 15(2 + n)² = 28(2 + n)(n) replace d with 2 + n

5n² + 60 + 60n + 15n² = 56n + 28n² multiply

20n² + 60n + 60 = 28n² + 56n combine like terms

28n² + 56n = 20n² + 60n + 60 just rearrange

-20n² -20n² subtract 20n² fr ea side
_______________________________
8n² + 56n = 60n + 60 subtract

-60n -60n subtract 60n fr ea side
_________________________________
8n² - 4n = 60 subtract

-60 - 60 subtract 60 fr ea side
_________________________________
8n² - 4n - 60 = 0 subtract
____ ___ ___ ___
4 4 4 4 divide thru by 4

2n² - n - 15 = 0 divide and cancel

(2n + 5)(n - 3) = 0 factor

n - 3 = 0 set this factor equal to 0

+ 3 +3 add 3 to each side
_______________
n = 3 add

d = 2 + 3 replace n with 3

d = 5 add

result: The fraction is 3/5 .

(9.) In a theater, the number of seats in each row is 16 fewer

than the number of rows. How many seats are there in

each row of a 1161 - seat theater?

s = r - 16

rs = 1161 here is the problem

r(r - 16) = 1161 replace s with r - 16

r² - 16r = 1161 multiply thru parentheses

-1161 -1161 subtract 1161 from each side
____________________
r² - 16r - 1161 = 0 subtract

(r - 43)(r + 27) = 0 factor

r - 43 = 0 set this factor equal to 0

+ 43 +43 add 43 to each side
________________
r = 43 add

s = 43 - 16 replace r with 43

s = 27 subtract

result: 27 seats

(10.) The length of a rectangle is 3 times its width. The

area of the rectangle is 192 square cm. What are the

dimensions of the rectangle?

L = 3w

wL = 192 here is the problem

w(3w) = 192 replace L with 3w

3w² = 192 multiply
___ _____
3 3 divide each side by 3

w² = 64 divide and cancel

w = 8 take square roots

L = 3(8) replace w with 8

L = 24 multiply

results: w = 8 and L = 24