2. Let n be an odd integer, and let 6, è € Zn. (a) Show that the set of all solutions to the equation x² +bx+c = Ō in Z, is -1 {(−6+6). 2-¹ | 8 is a square root of b² - 4c in Zn (i.e. 8² = 6² − 4c)}. (b) Find all solutions to each of the following equations. • x² + 3x + 5 = 0 in Z15 x² + 13x + 1 = 0 in Z23 • 2x² + x + 1 = 0 in Z15
2. Let n be an odd integer, and let 6, è € Zn. (a) Show that the set of all solutions to the equation x² +bx+c = Ō in Z, is -1 {(−6+6). 2-¹ | 8 is a square root of b² - 4c in Zn (i.e. 8² = 6² − 4c)}. (b) Find all solutions to each of the following equations. • x² + 3x + 5 = 0 in Z15 x² + 13x + 1 = 0 in Z23 • 2x² + x + 1 = 0 in Z15
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. Let n be an odd integer, and let b, c € Zn.
(a) Show that the set of all solutions to the equation x² + bx+c = Ō in Z₂ is
{(−5+6) · 2−¹ | 8 is a square root of b² − 4c in Zn (i.e. 8² = b² − 4c)}.
(b) Find all solutions to each of the following equations.
• x² + 3x + 5 = 0 in Z15
● x² + 13x + 1 = 0 in Z23
● 2x² + x + 1 = Ō in Z15
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