Rewrite x^2 - 8xy + 16y^2 as a perfect square using an algebraic identity.

Step-by-step solution

1. 1

   Decide which identity to test. A three-term expression with two perfect-square outer terms and a single middle term is a candidate for

   $a^2-2ab+b^2=(a-b)^2$

   Testing this first is faster than trial-and-error factoring, but the test has to be done properly: two squares are not enough, the middle term must match $-2ab$ exactly.

2. 2

   Identify $a$ and $b$ from the outer terms. The first term is already a square, $x^2$, so $a=x$. The last term is $16y^2=(4y)^2$, so $b=4y$. Note $b$ is the whole quantity $4y$, not just $y$ — carrying the coefficient inside $b$ is what makes the next check work.

3. 3

   Verify the middle term. With $a=x$ and $b=4y$,

   $-2ab=-2\cdot x\cdot 4y=-8xy$

   which is exactly the middle term given. Because all three terms match the pattern, the trinomial really is a perfect square. Had the middle term been anything else, say $-4xy$, the expression would not be a perfect square and would need a different method.

4. 4

   Write the square of the difference. The sign inside the bracket copies the sign of the middle term, so a $-8xy$ gives a minus:

   $x^2-8xy+16y^2=(x-4y)^2$

5. 5

   Check by expanding back. $(x-4y)^2=x^2-2(x)(4y)+16y^2=x^2-8xy+16y^2$. A numeric check is even quicker: at $x=5,\,y=1$ the original is $25-40+16=1$ and $(5-4)^2=1$. Both agree, so $(x-4y)^2$ is the answer.