1. MTH-4108-1
Chapter Review 9
Quadratic Functions
1. The difference between 2 numbers is 12. The product of the larger number increased
by 7 and the smaller number reduced by 9 is 128. What are the numbers?
Let (x) = smaller number
Let (x + 12) = LARGER number
−b± ∆
( x − 9)( x + 12 + 7 ) = 128 2a
( x − 9)( x + 19) = 128 − 10 ± 100 − 4(1)( − 299)
Zeros: 2
x 2 + 10 x − 171 − 128 = 0
− 10 ± 1296
x 2 + 10 x − 299 = 0
2
{ − 23,13}
13 + 12 = 25
ANS: The numbers are 13 and 25.
2. The sum of two numbers is 14. Two-thirds of the larger number increased by 9
multiplied by the smaller number less 2 is 45. What are the numbers?
Let (x) = LARGER number
Let (14 – x) = smaller number
( 2 3 x + 9)(14 − x − 2) = 45 −b± ∆
( 2 3 x + 9)( − x + 12) = 45
2a
− 1.5 ± 2.25 − 4(1)( − 94.5)
− 2 x 2 − 9 x + 8 x + 108 − 45 = 0 Zeros: 2
3
− 1.5 ± 380.25
− 2 x 2 − x + 63 = 0
3 2
x + 1.5 x − 94.5 = 0
2
{ − 10.5,9}
14 – 9 = 5
ANS: The numbers are 9 and 5.
3. The product of twice a number and half the same number less 1 is 288. What is
the number?
Let (x) = the number
−b± ∆
2a
2 ± 4 − 4(1)( − 288)
( 2
)
2 x 1 x − 1 = 288
Zeros: 2(1)
x − 2 x − 288 = 0
2
2 ± 1156
2
{ − 16,18}
ANS: The number is 18.

2. 4. The sum of the squares of 3 consecutive numbers is 434. What are the numbers?
Let (x) = 1st number
Let (x + 1) = 2nd number
Let (x +2) = 3rd number
−b± ∆
2a
x 2 + ( x + 1) + ( x + 2) = 434
2 2
− 2 ± 4 − 4(1)( − 143)
x 2 + x 2 + 2 x + 1 + x 2 + 4 x + 4 − 434 = 0
Zeros: 2
3 x 2 + 6 x − 429 = 0
− 2 ± 576
x 2 + 2 x − 143 = 0 2
{ − 13,11}
11 + 1 = 12
11 + 2 = 13
ANS: The numbers are 11, 12 and 13.
5. The sum of the squares of 3 consecutive odd numbers is 515. What are the
numbers?
Let (x) = 1st odd number
Let (x + 2) = 2nd odd number
Let (x +4) = 3rd odd number
−b± ∆
2a
x 2 + ( x + 2) + ( x + 4 ) = 515
2 2
− 4 ± 16 − 4(1)( − 165)
x 2 + x 2 + 4 x + 4 + x 2 + 8 x + 16 − 515 = 0
Zeros: 2(1)
3 x 2 + 12 x − 495 = 0
− 4 ± 676
x 2 + 4 x − 165 = 0
2
{ − 5,11}
11 + 2 = 13
11 + 4 = 15
ANS: The numbers are 11, 13 and 15.

3. 6. The difference between two numbers is 23. Five-sixths of the larger number less
20 multiplied by the smaller number increased by one is 800. What are the
numbers?
Let (x) = smaller number
Let (23 + x) =LARGER number
( x + 1) [ 5 6 ( 23 + x ) − 20] = 800
( x + 1) ( 5 6 x + 115 6 − 120 6 ) = 800 −b± ∆
2a
( x + 1) ( 5 6 x − 5 6 ) = 800 0 ± 0 − 4(1)( − 961)
5
6
( x + 1)( x − 1) = 800 Zeros: 2
± 3844
( x + 1)( x − 1) = 800 × 6 5 2
x 2 − 1 = 960 { − 31,31}
x 2 − 961 = 0
31 + 23 = 54
ANS: The numbers are 31 and 54.
7. The sum of two numbers is 38. Double the one number less 2 multiplied by half
the 2nd number less 1 is 300. What are the numbers?
Let (x) = LARGER number
Let (38 – x) = smaller number
( 1 2 x − 1)[ 2( 38 − x ) − 2] = 300 −b± ∆
2a
( 1 2 x − 1)( 76 − 2 x − 2) = 300 39 ± 1521 − 4(1)( 374 )
( 1 2 x − 1)( − 2 x + 74) = 300 Zeros: 2
39 ± 25
− 1x 2 + 37 x + 2 x − 74 − 300 = 0
2
x 2 − 39 x + 374 = 0 { − 17,22}
38 – 22 = 16
ANS: The numbers are 16 and 22.

4. 8. The product of one third a number less 6 and half the same number increased by
four is 720. What is the number?
Let (x) = the number
−b± ∆
( 1 2 x + 4)( 13 x − 6) = 720 2a
1 x 2 − 3 x + 4 x − 24 − 720 = 0 10 ± 100 − 4(1)( − 4464 )
6 3 Zeros: 2(1)
1 x 2 − 5 x − 744 = 0
6 3 10 ± 17956
x − 10 x − 4464 = 0
2
2
{ − 62,72}
ANS: The number is 72.
9. The sum of the squares of three even consecutive numbers is 596. What are the
numbers?
Let (x) = 1st even number
Let (x + 2) = 2nd even number
Let (x +4) = 3rd even number
−b± ∆
2a
x 2 + ( x + 2) + ( x + 4 ) = 596
2 2
− 4 ± 16 − 4(1)( − 192)
x 2 + x 2 + 4 x + 4 + x 2 + 8 x + 16 − 596 = 0
Zeros: 2(1)
3 x 2 + 12 x − 576 = 0
− 4 ± 676
x 2 + 4 x − 192 = 0
2
{ − 16,12}
12 + 2 = 14
12 + 4 = 16
ANS: The numbers are 12, 14 and 16.
10. The sum of the squares of three consecutive numbers is 869. What are the 3
numbers?
Let (x) = 1st number
Let (x + 1) = 2nd number
Let (x +2) = 3rd number
−b± ∆
2a
x 2 + ( x + 1) + ( x + 2) = 869
2 2
− 2 ± 4 − 4(1)( − 288)
x 2 + x 2 + 2 x + 1 + x 2 + 4 x + 4 − 869 = 0
Zeros: 2
3 x 2 + 6 x − 864 = 0
− 2 ± 1156
x 2 + 2 x − 288 = 0 2
{ − 18,16}
16 + 1 = 17
16 + 2 = 18
ANS: The numbers are 16, 17 and 18.