Question 59
The calculator $MK$-2020 can perform the following three operations on numbers in its memory:
1) Determine whether two chosen numbers are equal.
2) Add two chosen numbers.
3) For chosen numbers $a$ and $b$, determine the real roots of the equation $x^{2} + a x + b = 0$ or announce that the equation has no real roots.
The results of each operation are stored in its memory. Initially, the memory contains only the number $z$, which is unknown to its users.
a) Using these operations, determine how to find out whether $z$ is nonzero.
b) Find the roots of the quadratic equation in $x$:
$$
x^{2} + 2z x + z = 0
$$
c) How can we use the $MK$-2020 calculator to determine whether $z$ equals $1$?
1) Determine whether two chosen numbers are equal.
2) Add two chosen numbers.
3) For chosen numbers $a$ and $b$, determine the real roots of the equation $x^{2} + a x + b = 0$ or announce that the equation has no real roots.
The results of each operation are stored in its memory. Initially, the memory contains only the number $z$, which is unknown to its users.
a) Using these operations, determine how to find out whether $z$ is nonzero.
b) Find the roots of the quadratic equation in $x$:
$$
x^{2} + 2z x + z = 0
$$
c) How can we use the $MK$-2020 calculator to determine whether $z$ equals $1$?