Question

find the products ,,,, (a/2 + 4b/3) (a/2 + 4b/3) image explanation NOTICE: those who will help me in finding this question answer I'll mark them as brainliest​

Answer 1

Answer:

I was wondering if someone could take the time to look over this proof and make sure it is correct. I greatly appreciate the help.

Proposition: If a and b are integers, then a2−4b−3≠0.

Proof: Assume a,b∈Z and, for contradiction's sake, a2−4b−3=0. Solving for a2, we find a2=4b+3. Clearly, a2≡3(mod 4).

Now, we can factor 2 out of the left-hand side of a2=4b+3 yielding a2=2(2b+1)+1. Thus, by the definition of odd, a2 is odd. Since a2 is odd, a must be odd. By the definition of odd, we can write a=2c+1 where c∈Z.

Now we can substitute for a in a2 to find a2=(2c+1)2=4c2+4c+1. Factoring 4 out from the first two terms, we discover a2=4(c2+c)+1. Clearly, a2≡1(mod 4). Earlier, however, we found that a2≡3(mod 4). Since a can not be congruent to both 1 and 3 modulo 4, we have a contradiction. Therefore, if a,b∈Z, then a2−4b−4≠0

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