rbtta23 wrote:
If x is an integer, which of the following must be an even integer?
A: x^2-x-1
B: x^2-4x+6
C: x^2-5x+5
D: x^2+3x+8
E: x^2+2x+10
We first must recognize that when squaring an odd integer, the result will still be odd and when squaring an even integer, the result will still be even. That fact will help when analyzing the answer choices.
It will also be helpful to use our addition, subtraction, and multiplication rules, for odd and even numbers, when analyzing the answer choices.
Addition RulesE + E = Even
O + O = Even
E + O = Odd
*the same rules apply for subtraction
Multiplication RulesE x E = Even
E x O = Even
O x O = Odd
Now let's analyze each answer choice for when x is odd and when x is even.
A) x^2-x-1When x is even: E - E - 1 = odd
Since we have found an odd result, answer A is not correct.
B) x^2 - 4x + 6When x is even: E - E + 6 = even
When x is odd: O - E + 6 = odd
Since we have found an odd result, answer B is not correct.
C) x^2 - 5x + 5When x is even: E - E + 5 = odd
Since we have found an odd result, answer C is not correct.
D) x^2 + 3x + 8When x is even: E - E + 8 = even
When x is odd: O + O + 8 = even
Since we only get an even result,
answer D is correct.
E) x^2 + 2x + 10When x is even: E + E + 10 = even
When x is odd: O + E + 10 = odd
Since we have found an odd result, answer E is not correct.
Answer: D
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