Question

Let $f\left(x\right)=a{x}^2+bx+c$where $a,b,c$are odd integers. If x is an integer, show that $f\left(x\right)$must be an odd integer.

Short answer

We showed that if $x$is an integer then$f\left(x\right)$must be odd

Step-by-step solution

Given information

We are given a quadratic equation $f\left(x\right)=a{x}^2+bx+c$where $a,b,c$are odd.

Checking for even or odd

When $x$is even

We have,

$$\begin{gathered} f\left(x\right)=a{x}^2+bx+c \\ f\left(x\right)=\left(odd\right){\left(even\right)}^2+\left(odd\right)\left(even\right)+\left(odd\right) \\ f\left(x\right)=odd \end{gathered}$$

When x is odd

role="math" localid="1646066015968" $$\begin{gathered} f\left(x\right)=a{x}^2+bx+c \\ f\left(x\right)=\left(odd\right){\left(odd\right)}^2+\left(odd\right)\left(odd\right)+\left(odd\right) \\ f\left(x\right)=odd \end{gathered}$$

Conclusion

We proved that when $f\left(x\right)=a{x}^2+bx+c$where $a,b,c$are odd then if $x$is a integer then $f\left(x\right)$is a odd