Squaring a binomial with a subtraction term

Key takeaway: This problem tests binomial expansion and the special square pattern. The important detail is that the middle term must be doubled, and the constant term must stay positive after squaring.

Key idea

Use the identity:

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

For this expression, let $a=y$ and $b=8$.

Step-by-step method

Substitute into the formula:

$(y-8)^2 = y^2 - 2(y)(8) + 8^2$

Now simplify each part:

$y^2 - 16y + 64$

So the correct expansion is:

$\boxed{y^2 - 16y + 64}$

Why the middle term is negative

The sign in the middle comes from the subtraction inside the binomial. When you square a difference, the middle term becomes negative because of the $-2ab$ rule.

The last term is always positive, because a negative times a negative becomes positive:

$(-8)^2=64$

Common mistakes

A very common error is writing $y^2 - 16y - 64$. The last term should not be negative. Another error is forgetting to double the product of the two terms, which leads to $y^2 - 8y + 64$ instead of the correct expansion.

Final check

The perfect square trinomial pattern gives a quick way to confirm the answer without doing full distribution. The result is $y^2 - 16y + 64$.

---

Pitfalls the pros know 👇 The biggest trap is sign handling. Students often remember that the binomial is subtracting 8, but then they incorrectly make the constant term negative as well. Squaring always makes the last term positive because $(-8)^2=64$. Another common slip is forgetting the factor of 2 in the middle term. If you only multiply $y$ by 8 once, the expansion will be incomplete. For binomial squares, check the first term, the doubled product, and the square of the second term every time.

What if the problem changes? If the expression were $(y+8)^2$, the expansion would be $y^2+16y+64$ because the middle term changes sign. If it were $(y-5)^2$, the result would be $y^2-10y+25$. These variants show the same perfect square pattern: the first and last terms are squares, and the middle term is twice the product with the appropriate sign.

Tags: perfect square trinomial, binomial expansion, middle term coefficient