(a) Prove that Vr € R, ze Q⇒ (x + 1) E Q. Hint: go back to the definition of Q, and show that if a satisfies that definition, then so does (x + 1). (b) Prove that Vr € R, z Q⇒ (x + 1) # Q. Hint: can you prove the contrapositive of this (c) statement? Prove that Vr € R, z² + 6x +8.5 = 0⇒zQ. Hint: you may want to use results that are similar to 2a and 2b. C
(a) Prove that Vr € R, ze Q⇒ (x + 1) E Q. Hint: go back to the definition of Q, and show that if a satisfies that definition, then so does (x + 1). (b) Prove that Vr € R, z Q⇒ (x + 1) # Q. Hint: can you prove the contrapositive of this (c) statement? Prove that Vr € R, z² + 6x +8.5 = 0⇒zQ. Hint: you may want to use results that are similar to 2a and 2b. C
Intermediate Algebra
10th Edition
ISBN:9781285195728
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter6: Quadratic Equations And Inequalities
Section6.4: Quadratric Formula
Problem 61PS
Related questions
Question
Pleaseh help me with these question. I am having trouble understanding what to do
Please show all work
Thank you
![2. For this question, you will prove that the roots of z² + 6x + 8.5 are irrational. In other words, that
Va € R, z² + 6x + 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-√b²-4ac,
-b+√b²-4ac
Vr € R, (ar²+bx+c=0) ^ (b²-4ac > 0) ⇒ (z =
-) V (x=
2a
2a
Please write complete proofs in each subquestion, without referring back to earlier subquestions.
1
(a)
Prove that VI ER, re Q⇒ (r+1) EQ. Hint: go back to the definition of Q, and show
that if a satisfies that definition, then so does (x + 1).
(b)
Prove that Vr € R, z Q⇒ (x + 1) #Q. Hint: can you prove the contrapositive of this
statement?
(c)
Prove that Vr € R, z² + 6x +8.5 = 0⇒xQ. Hint: you may want to use results that
are similar to 2a and 2b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa61a78e1-5493-43a3-9d06-1e916681d4a2%2Faa8a0389-0e27-4305-b1fc-3367e5f9b4ff%2Fz049xv9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. For this question, you will prove that the roots of z² + 6x + 8.5 are irrational. In other words, that
Va € R, z² + 6x + 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-√b²-4ac,
-b+√b²-4ac
Vr € R, (ar²+bx+c=0) ^ (b²-4ac > 0) ⇒ (z =
-) V (x=
2a
2a
Please write complete proofs in each subquestion, without referring back to earlier subquestions.
1
(a)
Prove that VI ER, re Q⇒ (r+1) EQ. Hint: go back to the definition of Q, and show
that if a satisfies that definition, then so does (x + 1).
(b)
Prove that Vr € R, z Q⇒ (x + 1) #Q. Hint: can you prove the contrapositive of this
statement?
(c)
Prove that Vr € R, z² + 6x +8.5 = 0⇒xQ. Hint: you may want to use results that
are similar to 2a and 2b.
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