Question

I f 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

Answer 1

d

Answer 2

why though?

Answer 3

odd * odd = odd
even * even = even
even +even=even
odd+odd =even

Answer 4

for x^2+3x+8
is x is odd then x^2 is odd and 3x is odd
odd+odd+even=even

if x is even then x^2 is even and 3x is even
even+even+even=even

Answer 5

you could also use x=2k+1 and 2k and see that both are even in (d)

Answer 6

if x is even, then x=2n for some integer n
so x^2-x-1=(2n)^2-2n-1=(2n)^2+(2n+1)=2(2n^2)+(2n+1)=even+odd=even
but what if x is odd, then x=2n+1 for some integer n
so x^2-x-1=(2n+1)^2-(2n+1)-1=(2n+1)(2n+1-1)-1=(2n+1)(2n)-1=2n(2n+1)-1
=even-1=odd

Answer 7

so choice a it like totally not right since we got an odd output

Answer 8

let x=0 to see (a) does not work

Answer 9

or x=3
9-3-1=6-1=5

Answer 10

but 0 is pretty

Answer 11

prettier*

Answer 12

thanks guys