Prove each statement by contradiction.ExerciseFor all odd integers a, b, and c, if z is a...

Prove each statement by contradiction.

Exercise

For all odd integers a, b, and c, if z is a solution of ax2 + bx + c = 0 then z is irrational. (In the proof, use the properties of even and odd integers that are listed in Example.)

Example

Deriving Additional Results about Even and Odd Integers

Suppose that you have already proved the following properties of even and odd integers:

1. The sum, product, and difference of any two even integers are even.

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2. The sum and difference of any two odd integers are even.

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3. The product of any two odd integers is odd.

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4. The product of any even integer and any odd integer is even.

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5. The sum of any odd integer and any even integer is odd.

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6. The difference of any odd integer minus any even integer is odd.

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7. The difference of any even integer minus any odd integer is odd.

Use the properties listed above to prove that if a is any even integer and b is any odd integer, then  is an integer.

Solution

Suppose a is any even integer and b is any odd integer. By property 3, b2 is odd, and by property 1, a2 is even. Then by property 5, a2 + b2 is odd, and because 1 is also odd, the sum (a2 + b2)+ 1 = a2 + b2 + 1 is even by property 2. Hence, by definition of even, there exists an integer k such that a2 + b2 + 1 = 2k. Dividing both sides by 2 gives , which is an integer. Thus  is an integer [as was to be shown].