[quadratic word problems][section 48]

(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
____________
n = 6 add

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
______________
w = 24 add

L = 24 + 20 replace w with 24

L = 44 add

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
_________________
w = 18 add

L = 40 - 18 replace w with 18

L = 22 subtract

result: w = 18 and L = 22