For this question, you will prove that the roots of z² +6z+8.5 are irrational. In other words, that Vz € R, z² +6+8.5=0->zgQ. 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-√²-4ac -b+√²-4ac 2a 2a Vz € R₁ (az²+bx+c=0) ^ (6² - 4ac ≥ 0) (z= 5) V (z Please write complete proofs in each subquestion, without referring back to earlier subquestions. (a) Prove that Vr € R, z € Q (z+1) € Q. Hint: go back to the definition of Q, and shou that if z satisfies that definition, then so does (z + 1). (b) Prove that Vr € R,r Q (z+1) g Q. Hint: can you prove the contrapositive of this statement? (c) Prove that Vz € R, z² +6z+8.5=0 zgQ. Hint: you may want to use results that are similar to 20 and 26.
For this question, you will prove that the roots of z² +6z+8.5 are irrational. In other words, that Vz € R, z² +6+8.5=0->zgQ. 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-√²-4ac -b+√²-4ac 2a 2a Vz € R₁ (az²+bx+c=0) ^ (6² - 4ac ≥ 0) (z= 5) V (z Please write complete proofs in each subquestion, without referring back to earlier subquestions. (a) Prove that Vr € R, z € Q (z+1) € Q. Hint: go back to the definition of Q, and shou that if z satisfies that definition, then so does (z + 1). (b) Prove that Vr € R,r Q (z+1) g Q. Hint: can you prove the contrapositive of this statement? (c) Prove that Vz € R, z² +6z+8.5=0 zgQ. Hint: you may want to use results that are similar to 20 and 26.
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. For this question, you will prove that the roots of z² +6r+8.5 are irrational. In other words, that
Vr € R, z² + 6 + 8.5=0=zQ.
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- -4ac
4ac > 0) ⇒ (x=
-b+√b² - 4ac
2a
Vr € R, (ar²+bx+c=0) ^ (6²-4ac
:) V (z =
2a
Please write complete proofs in each subquestion, without referring back to earlier subquestions.
1
(a)
Prove that Vr € R, z € Q⇒ (z+1) € Q. Hint: go back to the definition of Q, and show
that if a satisfies that definition, then so does (2+1).
(b)
Prove that Vr € R, z Q➡ (z+1) Q. Hint: can you prove the contrapositive of this
statement?
(c)
Prove that VI ER,z² +6z+8.5=0⇒zQ. Hint: you may want to use results that
are similar to 2a and 26.
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