(1.)  The sum of two numbers is 12.  The sum of their squares

is 74.  What are the numbers?

x + y = 12

x2 + y2 = 74            here is the problem

x + y = 12

-x        -  x       subtract x from each side
_____________________
y = 12 - x          subtract

x2 + (12 - x)2 = 74       replace y with 12 - x

x2 + 144 - 24x + x2 = 74   square the binomial

2x2 - 24x + 144 = 74           combine like terms

-    74   -74      subtract 74 from each side
_______________________
2x2 - 24x + 70 = 0           subtract
___   ___  ___   ___
2      2    2    2     divide thru by 2

x2 - 12x + 35 = 0         divide

(x - 5)(x - 7) = 0      factor

x - 5 = 0       x - 7 = 0     set each factor equal to 0

+  5  +5          + 7  +7     add this to each side
______________ _________________
x = 5  ,     x =    7     add

y = 12 - 5         y = 12 - 7        replace x with 5 & 7

y = 7             y = 5               subtract

result: The numbers are 5 and 7.

(2.)   The difference of two number is 3.   The sum of their

squares is 117.   What are the numbers?

x - y = 3

x2 + y2 = 117               here is the problem

x - y = 3                  put this here

+y  +y              add y to each side
_____________
x = y + 3               add

(y + 3)2 + y2 = 117        replace x with y + 3

y2 + 6y + 9 + y2 = 117        square the binomial

2y2 + 6y + 9 = 117          combine like terms

-117   -117       subtract 117 fr ea side
___________________________
2y2 + 6y - 108 = 0            subtract
____  ___  ____ ____
2      2   2    2     divide thru by 2

y2 + 3y - 54 = 0          divide and cancel

(y + 9)(y - 6) = 0          factor

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

-9 -9         +6 +6   add this to each side
_____________   ___________
y = -9   ,     y = 6       add

x =  -9 + 3       x = 6 + 3    replace y with -9 and 6

x = -6          x = 9               add

results:  x = -6 and y = -9  or  x = 9 and y = 6

(3.)    Find two consecutive integers whose product is 210.

n(n + 1) = 210           here is the problem

n2 + n = 210         multiply thru parentheses

-210   -210         subtract 210 from each side
_________________
n2 + n - 210 = 0        subtract

(n + 15)(n - 14) = 0        factor

n + 15 = 0     n - 14 = 0     set each factor equal to 0

-15  -15    +  14 14     add this to each side
_____________ _____________
n = -15    ,    n = 14      add

results:  The numbers are -15 and -14.  Or, 14 and 15.

(4.)   n(n + 2) = 80             here is the problem

n2 + 2n = 80        multiply thru parentheses

-80  -80         subtract 80 from each side
__________________

n2 + 2n - 80 = 0          subtract

(n + 10)(n - 8) = 0           factor

n + 10 = 0    n - 8 = 0    set each factor equal to 0

-10  -10      + 8  +8   add this to each side
_____________ ____________
n = -10   ,    n = 8        add

results:  The numbers are -10 and -8, or 8 and 10.

(5.)  Find two consecutive negative integers such that the

sum of their squares is 113.

n2 + (n + 1)2 = 113          here is the problem

n2 + n2 + 2n + 1 = 113     square the binomial

2n2 + 2n + 1 = 113        combine like terms

- 113  -113      subtract 113 from each side
________________________
2n2 + 2n - 112 = 0       subtract
____  ___  ____  ___
2   2    2     2    divide thru by 2

n2 + n - 56 = 0       divide and cancel

(n + 8)(n - 7) = 0     factor

n + 8 = 0      set this factor equal to 0

-8  -8      subtract 8 from each side
______________

n = -8       subtract

result:  the numbers are -8 and -7 .

(6.)  The square of a positive integer is 30 more than

the integer.   Find the integer.

n2 = n + 30             here is the problem

-n  -n           subtract n from each side
_____________________
n2 - n =    30       subtract

-30  -30  subtract 30 from each side
_________________
n2 - n - 30 = 0          subtract

(n + 5)(n - 6) = 0          factor

n - 6 = 0         set this factor equal to 0

+6   +6           add 6 to each side
____________

result:  The integer is 6.

(7.)  Twice the square of a number is 144 more than twice the

number.   What is the number?

2n2 = 2n + 144           here is the problem
___  ___  ____
2   2    2         divide thru by 2

n2 = n + 72             divide and cancel

-n -n            subtract n from each side
_______________
n2 - n   =  72          subtract

-72  -72         subtract 72 from each side
_________________
n2 - n - 72 = 0             subtract

(n - 9)(n + 8) = 0            factor

n - 9 = 0     n + 8 = 0   set each factor equal to 0

+ 9  +9       -   8  -8   add this to each side
_____________ ______________
n = 9     ,     n = -8   add

results:  n = 9  or n = -8

(8.)  A rectangular parking lot is 20 meters longer than it is

wide.   The area of the lot is 1056 square meters. Find

the dimensions of the lot.

L = w + 20

wL = 1056                    here is the problem

w(w + 20) = 1056          replace L with w + 20

w2 + 20w = 1056          multiply thru parentheses

-1056  -1056        subtract 1056 from each side
_____________________
w2 + 20w - 1056 = 0          subtract

(w + 44)(w - 24) = 0          factor

w - 24 = 0    set this factor equal to 0

+ 24  +24         add 24 to each side
______________

L = 24 + 20            replace w with 24

result:  w = 24 and L = 44

(9.)   Find two consecutive odd integers such that the square

of the second number decreased by  the first number

is 14.

(n + 2)2 - n = 14           here is the problem

n2 + 4n + 4 - n = 14      square the binomial

n2 + 3n + 4 = 14             combine like terms

-14  -14        subtract 14 from each side
____________________
n2 + 3n - 10 = 0            subtract

(n + 5)(n - 2) = 0          factor

n + 5 = 0      set this factor equal to 0

-5  -5      subtract 5 from each side
______________
n     = -5       subtract

result:  The numbers are -5 and -3.

(10.)  The perimter of a picture frame is 80 cm.   The area

is 396 square cm.   Find the dimensions of the frace.

2L + 2w = 80

wL = 396                   here is the problem

2L + 2w = 80
___  ___  ____
2   2      2     divide thru by 2

L + w = 40           divide and cancel

- w    -   w          subtract w from each side
_______________________
L        = 40 - w          subtract

w(40 - w) = 396       replace L with 40 - w

40w - w2 = 396          multiply thru parentheses

-40w + w2 = -396        multiply thru by -1

w2 - 40w = -396            rearrange terms

+ 396  +396        add 396 to each side
______________________
w2 - 40w + 396 = 0           add

(w - 22)(w - 18) = 0      factor

w - 18 = 0             set this factor equal to 0

+ 18  +18         add 18 to each side
_________________