[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
______________

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
_______________

L = 10 + 2(20)    replace w with 20

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
________________

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
_______________

d = 2 + 3              replace n with 3

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
________________

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