Download Finding Complex Solutions of Quadratic Equations

Name ________________________________________ Date __________________ Class __________________
LESSON
4-3
Finding Complex Solutions of Quadratic Equations
Practice and Problem Solving: C
Solve using the quadratic formula.
1. 2 x 2 − 6 x − 1 = 0
2. x 2 − x = −12
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3. −2 x 2 = 5 x − 20
4. −4 x 2 − 3 x − 36 = 0
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Find the discriminant of each equation. Then determine the number of
real or nonreal solutions.
5. 2 x 2 + 7 = −4 x
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6. x 2 − 3 = −6 x
7. 4 x 2 + 4 = −8 x
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Complete the square for each expression. Write the resulting
expression as a binomial squared.
8. x2− 22x +
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9. x2 + 9x +
10. 64x2− 48x +
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Solve each equation by completing the square.
11. 14 x + x 2 = 24
12. 2 x 2 − 8 x = −2
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13. x 2 = 3 x + 4
14. 4 x 2 + 32 x + 16 = 0
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15. A pedestrian suspension bridge built above a road is supported by a
parabolic arch. The height in feet of the arch is given by the equation
h(x) = x(13.5 − x). Can a semi-truck with a height of 31.25 feet pass
under the highest point of the arch? Use the discriminant to explain.
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Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
65
7. 0; one real solution
Practice and Problem Solving:
Modified
8. 4; ( x − 2 )
1. −1 for all blanks
2
9. 36; ( x + 6 )
2. −1 for all blanks
3. −1 for all blanks
10. 1; ( 5 x − 1)
4. (2i + 9i ) + 6
11. x = −3, 1
5. (3i − 7i ) − 4
12. x =
6. [−5 + (−2)] + (2i + 8i )
7. 3, 3, −2i, −2i;−12i2; −1; 20, 12; 20, 9
2
2
13. x = 3 ± 19
14. x = −
2
9. i, i ; i ; i ; i; 1
3
13
±
2
2
15. No; because the discriminant is negative,
the equation has no real solutions, so the
baseball will never reach the height of the
roof.
Reading Strategies
1. Sum: (6 − 9x) + (−3 + 4x) = (6 + (−3)) +
(−9x + 4x) = 3 + (−5x) = 3 − 5x
Difference: (6 − 9x) − (−3 + 4x) =
(6 − (−3)) − (−9x − 4x) = 9 − (−13x) =
9 + 13x Product: (6 − 9x)(−3 + 4x) = −18 +
24x + 27x − 36x2 = −18 + 51x − 36x2
Practice and Problem Solving: C
2. Sum: (6 − 9i ) + (−3 + 4i ) = (6 + (−3)) +
(−9i + 4i ) = 3 + (−5i ) = 3 − 5i
Difference: (6 − 9i ) − (−3 + 4i ) = (6 − (−3)) −
(−9i − 4i ) = 9 + (−13i ) = 9 − 13i
Product: (6 − 9i )(−3 + 4i ) = −18 + 24i + 27i
− 36i 2 = −18 + 51i − 36(−1) = 18 + 51i
3. Multiplication; Possible explanation: The
product is different because i2 can be
simplified to −1, but x2 cannot be further
simplified.
1. x =
3 ± 11
2
2. x =
1 ± i 47
2
3. x = −
5 ± 185
4
4. x = −
3 9i 7
±
8
8
5. −40; two nonreal solutions
6. 48; two real solutions
Success for English Learners
7. 0; one real solution
8. 121; ( x − 11)
1. No, −n is a real number only if n is a
negative number.
9.
LESSON 4.3
Practice and Problem Solving: A/B
81
;
4
9⎞
⎛
⎜x + 2⎟
⎝
⎠
10. 9; ( 8 x − 3 )
1. x = −9, − 1
2
2
2
11. x = −7 ± 73
2. x = −1 ± i 3
12. x = 2 ± 3
−5 ± 37
3. x =
2
4. x = −
2
5
41
±
2
2
2
8. 3, 3, 2, 4i; i ; 14, i ; −4; 4; 4
2
2
13. x = −1, 4
14. x = −4 ± 2 3
7 i 31
±
4
4
15. Yes; because the discriminant is positive,
the equation has two real solutions, so the
semitruck can pass under the arch.
5. −23; two nonreal solutions
6. 4; two real solutions
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
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