Which expression is equivalent to $(a-2 b)^2$ ?
To find an expression equivalent to \((a-2b)^2\), we can use the algebraic identity for the square of a binomial difference, which states that \((x-y)^2 = x^2 - 2xy + y^2\). This identity allows us to expand expressions of the form (first term - second term) squared by squaring the first term, subtracting twice the product of the two terms, and adding the square of the second term.
We identify the terms in our expression \((a-2b)^2\). Comparing this to \((x-y)^2\), we have \(x = a\) and \(y = 2b\).
Applying the formula:
Now we combine these parts according to the identity:
\[(a-2b)^2 = a^2 - 2(a)(2b) + (2b)^2\]
\[(a-2b)^2 = a^2 - 4ab + 4b^2\]
Comparing our result with the provided options:
A: \(a^2 - 4b^2\)
B: \(a^2 + 4b^2\)
C: \(a^2 - 4a^2b^2 - 4b^2\)
D: \(a^2 - 4ab + 4b^2\)
Our calculated expression matches option D.
Expanding binomials is a fundamental skill in algebra used for simplifying complex equations in physics and engineering. For example, if you are calculating the area of a square-shaped field where each side length is reduced by a certain amount (represented as 2b), the total area can be determined by expanding the expression \((a-2b)^2\). This helps in understanding how changes in dimensions affect physical space or financial budgets efficiently.