Question

why is 4x^2-16x+16 a perfect square trinomial

Answer 1

To determine whether the expression \( 4x^2 - 16x + 16 \) is a perfect square trinomial, we need to recall that a perfect square trinomial is of the form \( (ax + b)^2 \), which expands to:

\[
a^2x^2 + 2abx + b^2
\]

To see if \( 4x^2 - 16x + 16 \) fits this form, we can analyze the quadratic expression.

1. **Identify \(a^2\), \(2ab\), and \(b^2\)**:
- The term \(4x^2\) suggests that \(a^2 = 4\), so \(a = 2\).
- The term \(16\) can be seen as \(b^2\), which suggests \(b = 4\) (since \((4)^2 = 16\)).
- Now we check the middle term \(2ab\):
\[
2ab = 2(2)(4) = 16
\]
Since the middle term in our given expression is \(-16x\), we realize that it should be \(-16\) instead of \(16\). This suggests \(b = -4\) (since \(b\) must be negative to produce a negative middle term).

2. **Write the expression as a square**:
With \(a = 2\) and \(b = -4\), we can express \(4x^2 - 16x + 16\) as:
\[
(2x - 4)^2
\]

Now let's expand \( (2x - 4)^2 \) to verify:
\[
(2x - 4)(2x - 4) = 4x^2 - 8x - 8x + 16 = 4x^2 - 16x + 16
\]

Since we obtained the original expression from the expansion, we confirm that:

\[
4x^2 - 16x + 16 = (2x - 4)^2
\]

Therefore, \(4x^2 - 16x + 16\) is indeed a perfect square trinomial.