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Mth 4108-1 - chapter 9 (ans)
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  • 1. MTH-4108-1 Chapter Review 9 Quadratic Functions1. 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.