If 1 is a zero of the quadratic polynomial $a{x}^2+bx+c$, which relation must its coefficients satisfy?

A zero makes the polynomial vanish, so $p(1)=a(1{)}^2+b(1)+c=a+b+c=0$. The other options are the conditions for something else: $a-b+c=0$ says −1 is a zero, $c=0$ says 0 is a zero (since the product $\frac{c}{a}$ would be 0), and $b^2=4ac$ says the two zeroes are equal.