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What is the completely factored form of $x^2-16 x y+64 y^2$?


A$x y(x-16+64 y)$
B$x y(x+16+64 y)$
C$(x-8 y)(x-8 y)$
D$(x+8 y)(x+8 y)$

Verified Answer :

  • Recognize the expression \(x^2 - 16xy + 64y^2\) as a perfect square trinomial following the pattern \(a^2 - 2ab + b^2 = (a-b)^2\).
  • Identify \(a=x\) and \(b=8y\) by taking the square roots of the first and last terms.
  • Verify the middle term by calculating \(2ab = 2(x)(8y) = 16xy\).
  • Conclude that the factored form is \((x-8y)^2\), which is \(\boxed{(x - 8y)(x - 8y)}\).

Explanation

  1. Analyzing the Expression Structure

We are given the expression \(x^2 - 16xy + 64y^2\). Our objective is to factor this expression completely. We observe that this is a quadratic trinomial with two variables, \(x\) and \(y\). Specifically, it follows the form of a perfect square trinomial, \(a^2 - 2ab + b^2\), which factors as \((a - b)^2\).

  1. Identifying the Components of the Perfect Square

To use the identity \(a^2 - 2ab + b^2 = (a - b)^2\), we identify the components. Comparing \(x^2 - 16xy + 64y^2\) to \(a^2 - 2ab + b^2\), we can see that:

  • The first term \(a^2 = x^2\), which implies \(a = x\).
  • The last term \(b^2 = 64y^2\), which implies \(b = \sqrt{64y^2} = 8y\).
  1. Verifying the Middle Term

Now, we must verify that the middle term matches the identity \(2ab\). Given \(a = x\) and \(b = 8y\), the middle term of our identity should be:
\[2ab = 2(x)(8y) = 16xy\]
The middle term of our expression is \(-16xy\), which matches the \(-2ab\) form in \(a^2 - 2ab + b^2 = (a - b)^2\). Therefore, the expression is indeed a perfect square of the binomial \((x - 8y)\).

  1. Factoring the Expression

Since we have confirmed that the expression \(x^2 - 16xy + 64y^2\) is a perfect square trinomial, we can rewrite it as:
\[x^2 - 16xy + 64y^2 = (x - 8y)^2\]
The expression \((x - 8y)^2\) is equivalent to \((x - 8y)(x - 8y)\). Comparing this to the given options, we find that this matches the third option. Thus, the completely factored form is \((x - 8y)(x - 8y)\).

Examples

Factoring perfect square trinomials is a fundamental skill in algebra, useful for solving quadratic equations and simplifying complex expressions. For example, if you are designing a square garden with area \(x^2 - 16xy + 64y^2\) square meters, knowing the side length is \((x-8y)\) meters allows you to easily calculate perimeter or plan irrigation layouts efficiently. Understanding these patterns helps in various fields, from structural engineering to statistical analysis of data distributions.

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