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If $a$ and $b$ are non-zero integers such that $a^2 + b^2 - 4a - 2b = 0$ and $a^2 - b^2 = 0$ , which of the following could be the value of $(a - b)$ ?

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Transformations of linear functions

Full solution

Q. If

a

and

b

are non-zero integers such that

a^2 + b^2 - 4a - 2b = 0

and

a^2 - b^2 = 0

, which of the following could be the value of

(a - b)

?

Analyze Equations: Analyze the given equations. $\newline$ We are given two equations: $\newline$ $1$ . $a^2 + b^2 - 4a - 2b = 0$ $\newline$ $2$ . $a^2 - b^2 = 0$ $\newline$ We need to find the possible values of $(a-b)$ .
Solve Second Equation: Solve the second equation. $\newline$ The second equation is a difference of squares, which can be factored as: $\newline$ $(a + b)(a - b) = 0$ $\newline$ Since $a$ and $b$ are non-zero integers, $a + b$ cannot be zero, so we must have: $\newline$ $a - b = 0$
Determine Value of $(a−b)$ : Determine the value of $(a−b)$ . From Step $2$ , we found that $a - b = 0$ . This means that $a$ must be equal to $b$ .
Verify with First Equation: Verify the solution with the first equation. $\newline$ Since $a = b$ , we can substitute $b$ for $a$ in the first equation: $\newline$ $b^2 + b^2 − 4b − 2b = 0$ $\newline$ $2b^2 − 6b = 0$ $\newline$ $b(2b − 6) = 0$ $\newline$ Since $b$ is a non-zero integer, we cannot have $b = 0$ , so we must have: $\newline$ $2b − 6 = 0$ $\newline$ $b = 3$ $\newline$ Since $a = b$ , $a$ is also $b$ $2$ .
Calculate $(a−b)$ : Calculate the value of $(a−b)$ . $\newline$ Since $a = b = 3$ , the value of $(a−b)$ is: $\newline$ $(a−b) = 3 − 3 = 0$

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