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SpartanG345
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http://www.cs.odu.edu/~toida/nerzic/content/problem_solving/problem_solving.html
example 3
========================================= Quote
Problem: Given that a, b, and c are odd integers, prove that equation ax2 + bx + c = 0 can not have a rational root.
Understanding the Problem: This is a "prove" type problem.
The hypothesis is that a, b, and c are odd integers, and the conclusion is that equation ax2 + bx + c = 0 can not have a rational root.
The hypothesis is straightforward. In the conclusion, "rational root" means a root, that is, the value of x that satisfies the equation, and that can be expressed as m/n, where m and n are integers. So the conclusion means that there is no number of the form m/n that satisfies the equation under the hypothesis.
Devising a Solution Plan: For this problem, let us try "proof by contradiction". When you are asked to prove the impossibility of an event or non-existence of certain things, this approach often is quite helpful.
Following the "proof by contradiction", let us assume that the conclusion is false, that is the equation ax2 + bx + c = 0 has a rational root m/n, where m and n are integers, when a, b, and c are odd integers. We can assume without loss of generality that m and n do not have any factors in common. Then
a(m/n)2 + b(m/n) + c = 0 . ------------------------ (1)
Let us try to derive a contradiction from this.
First let us make this equation simpler, that is, let us get rid of fractions.
Since n is not equal to 0, multiplying the both sides of (1) by n2, we get
am2 + bmn + cn2 = 0 . ------------------------ (2)
Since m is an integer, it is either even or odd. We are going to consider those cases one by one. That is "divide into cases".
Let us first consider the case when m is even.
Then n is odd, since otherwise m and n have a common factor 2. Now am2 + bmn is even, and cn2 is odd. Hence am2 + bmn + cn2 can not be 0.
Next let us consider the case when m is odd.
By an argument similar to the previous case, we can see that n is also odd.
If m and n are odd, then am2, bmn, and cn2 are all odd, since a, b, and c are odd integers. However, the sum of three odd numbers can not be equal to 0.
Thus by assuming that the conclusion is false, we have arrived at a contradiction, that is m/n does not satisfy the equation. Hence our assumption must be wrong, and therefore the conclusion is correct.
========================================================== End quote
I found this problem online, however i do not understand the last part, the one in bold. Would the last statement result in loss of generality.
Eg how would you get 3/2 this fits in the second case where m is odd. However in either case n is odd. so it is impossible to get 3/2...? If n is any even number then there can be common factors ie 9/6.
Any ideas?
example 3
========================================= Quote
Problem: Given that a, b, and c are odd integers, prove that equation ax2 + bx + c = 0 can not have a rational root.
Understanding the Problem: This is a "prove" type problem.
The hypothesis is that a, b, and c are odd integers, and the conclusion is that equation ax2 + bx + c = 0 can not have a rational root.
The hypothesis is straightforward. In the conclusion, "rational root" means a root, that is, the value of x that satisfies the equation, and that can be expressed as m/n, where m and n are integers. So the conclusion means that there is no number of the form m/n that satisfies the equation under the hypothesis.
Devising a Solution Plan: For this problem, let us try "proof by contradiction". When you are asked to prove the impossibility of an event or non-existence of certain things, this approach often is quite helpful.
Following the "proof by contradiction", let us assume that the conclusion is false, that is the equation ax2 + bx + c = 0 has a rational root m/n, where m and n are integers, when a, b, and c are odd integers. We can assume without loss of generality that m and n do not have any factors in common. Then
a(m/n)2 + b(m/n) + c = 0 . ------------------------ (1)
Let us try to derive a contradiction from this.
First let us make this equation simpler, that is, let us get rid of fractions.
Since n is not equal to 0, multiplying the both sides of (1) by n2, we get
am2 + bmn + cn2 = 0 . ------------------------ (2)
Since m is an integer, it is either even or odd. We are going to consider those cases one by one. That is "divide into cases".
Let us first consider the case when m is even.
Then n is odd, since otherwise m and n have a common factor 2. Now am2 + bmn is even, and cn2 is odd. Hence am2 + bmn + cn2 can not be 0.
Next let us consider the case when m is odd.
By an argument similar to the previous case, we can see that n is also odd.
If m and n are odd, then am2, bmn, and cn2 are all odd, since a, b, and c are odd integers. However, the sum of three odd numbers can not be equal to 0.
Thus by assuming that the conclusion is false, we have arrived at a contradiction, that is m/n does not satisfy the equation. Hence our assumption must be wrong, and therefore the conclusion is correct.
========================================================== End quote
I found this problem online, however i do not understand the last part, the one in bold. Would the last statement result in loss of generality.
Eg how would you get 3/2 this fits in the second case where m is odd. However in either case n is odd. so it is impossible to get 3/2...? If n is any even number then there can be common factors ie 9/6.
Any ideas?
