2. For this question, you will prove that the roots of r²+6x+8.5 are irrational. In other words, that VxR, x²+6x+8.5=0⇒2Q. You may use, without proof, the quadratic formula. In other words, you may use the fact that for all real a, b, and c, -b-√√b2-4ac 2a -b+√√b2-4ac -) V (x= 2a Vr € R, (ar² + bx + c = 0) A (62-4ac > 0) (1 = Please write complete proofs in each subquestion, without referring back to earlier subquestions. (a) (b) (c) Prove that VrЄR,Q(z+1) Q. Hint: go back to the definition of Q, and show that if x satisfies that definition, then so does (z+1). Prove that Vz ER, Q(x+1) Q. Hint: can you prove the contrapositive of this statement? Prove that VrЄR,²+6x+8.5=0&Q. Hint: you may want to use results that are simuar to 2a and 2b.
2. For this question, you will prove that the roots of r²+6x+8.5 are irrational. In other words, that VxR, x²+6x+8.5=0⇒2Q. You may use, without proof, the quadratic formula. In other words, you may use the fact that for all real a, b, and c, -b-√√b2-4ac 2a -b+√√b2-4ac -) V (x= 2a Vr € R, (ar² + bx + c = 0) A (62-4ac > 0) (1 = Please write complete proofs in each subquestion, without referring back to earlier subquestions. (a) (b) (c) Prove that VrЄR,Q(z+1) Q. Hint: go back to the definition of Q, and show that if x satisfies that definition, then so does (z+1). Prove that Vz ER, Q(x+1) Q. Hint: can you prove the contrapositive of this statement? Prove that VrЄR,²+6x+8.5=0&Q. Hint: you may want to use results that are simuar to 2a and 2b.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please help me with these questions. I a
Transcribed Image Text:2. For this question, you will prove that the roots of r²+6x+8.5 are irrational. In other words, that
VxR, x²+6x+8.5=0⇒2Q.
You may use, without proof, the quadratic formula. In other words, you may use the fact that for all
real a, b, and c,
-b-√√b2-4ac
2a
-b+√√b2-4ac
-) V (x=
2a
Vr € R, (ar² + bx + c = 0) A (62-4ac > 0) (1 =
Please write complete proofs in each subquestion, without referring back to earlier subquestions.
(a)
(b)
(c)
Prove that VrЄR,Q(z+1) Q. Hint: go back to the definition of Q, and show
that if x satisfies that definition, then so does (z+1).
Prove that Vz ER, Q(x+1) Q. Hint: can you prove the contrapositive of this
statement?
Prove that VrЄR,²+6x+8.5=0&Q. Hint: you may want to use results that
are simuar to 2a and 2b.
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