The Arithmetic Machine

Let the input be $X$ . Then machine will do processing like $2 X$ , then $2 X + 2$ and final output will be $(2 X + 2)^2$ .

Now both the outputs of Bob and Alice are $100$ . So, it simply means

$(2 X + 2)^2 = 100$

$4 (X + 1)^2 = 100$

$(X +1)^2 = 25$

$X + 1 = \pm 5$

$X = \pm 5 - 1 = \boxed 4$ or $\boxed {-6}$

Both $4$ and $-6$ would give $100$ as :

$4 \rightarrow 8 \rightarrow 10 \rightarrow 100$

$(-6) \rightarrow (-12) \rightarrow (-10) \rightarrow 100$

Let $x$ be any positive real number that results after multiplying by $2$ and adding $2$ , the first two operations. Observe that both $(x)^2$ and $(- x)^2$ will result in $x^2$ . Thus, the same number could have resulted from two different initial numbers entering the arithmetic machine.

In this case, Alice's number could have been $4$ , and Bob's $- 6$ .

Squaring can give the same answer, 100 for two different inputs (e.g. 10 and -10). Therefore, Alice could have inputted a number which resulted in 10 when it got to the squaring step and Bob could have inputted a number which resulted in -10 when it got to the squaring step, but they would both still get the same output.

if we input 4,

the result would be=(4 x 2 +2)^2=100

if, we put -6. the result would be= (-6 x 2 +2)^2=100.

so,it is not necessary for them to input the same digit

If the input is X, the arithmetic machine performs the equation (2x+2)^2=4x^2+8x+4. Since both Alice and Bob got 100 as an answer, this means that 4x^2+8x+4=100. Simplyfying, we get x^2+2x-24=0. Solving for x with the quadratic formula, with a=1, b=2 and c=-24, the answers are that x=-1+/-5, meaning that x=-6 and x=-4. Therefore, there are three possible scenarios: either both entered -6, both entered 4, or one entered -6 while the other entered 4.

$(2x+2)^2=100$ has two solutions, x=4 and x=-6

Let the input be $x$ , then we have:

$\begin{aligned} (2x+2)^2 & = 100 & \small \color{#3D99F6} \text{Taking the square root both sides } \\ 2x + 2 & = \pm 10 \\ \implies x & = \begin{cases} \dfrac {10-2}2 = 4 \\ \dfrac {-10-2}2 = -6 \end{cases} \end{aligned}$

No, not necessarily , as the input $x$ can take on two values.

What we need to remember is that basic arithmetic (addition, subtraction, multiplication and division) can only have one possible answer.

This means that in this case:

Multiply by 2
Add 2

Can only give one answer.

However, squaring is a bit more complicated. You can do:

$6\times6$

to get 36, but keep in mind the rule of multiplying negative numbers:

plus times plus equals plus
plus times minus equals minus
minus times plus equals minus
minus times minus equals plus

That last rule means that:

$-6\times-6$

also equals 36.

It should also be noted that squaring is multiplying a number by itself.

The machine evaluates a quadratic expression ( $2x + 2)^2$ .

The numbers entered in this case are the roots of a quadratic equation $x^2 + 2x - 24 = 0$ derived from $(2x + 2)^2 - 100 = 0$ . This expression can be factored as $(x - 4)(x + 6)$ showing that this equation has two distinct roots 4 and -6. Thus two different numbers entered can produce the same result.

Let the input number be x. Therefore, the final output is $(2x + 2)^{2}$ .

Therefore, $(2x + 2)^{2}=100$ .

Therefore, $2x + 2=+10$

OR

$2x + 2=-10$

Solving this gives us: $x = 4$ OR $x = -6$

So it is quite possible that Alice and Bob input different values.

Let the input be x, so the machine will multiply it 1st with 2 i.e. 2x then 2x+2, then (2x+2)^2... Now to get output as 100, (2x+2)^2 = 100 Hence we get x = 4 or -6

10 - 2 = 8 / 2 = 4 OR -10 - 2 = -12 /2 = -6 So the solution set is (-6,4)

Input could be a negative value or a positive value

$X \rightarrow 2X \rightarrow 2X + 2 \rightarrow (2X + 2)^2 = 4(X + 1)^2 = 100$

$\implies X^2 + 2X - 24 = 0 \implies X = -6,4$

100 as a result means that 10 OR -10 could have been squared to reach that point. Thus, the answers would be 4 and -6.

Enter the numbers 4 and -6 you would get 100 answer so it is not necessary that both have entered the same numbers..