Question

factor the expression
16x^2 - 40x + 25

Answer 1

To factor the expression \(16x^2 - 40x + 25\), we can start by observing whether it can be factored as a perfect square trinomial.

The general form of a perfect square trinomial is \( (ax + b)^2 = a^2x^2 + 2abx + b^2 \).

1. Identify \(a^2\) and \(b^2\):

- The coefficient of \(x^2\) is \(16\), so \(a^2 = 16\) which gives us \(a = 4\).
- The constant term is \(25\), so \(b^2 = 25\) which gives us \(b = 5\).

2. Check the middle term:

The middle term in the trinomial is \(-40x\).

- Calculate \(2ab\):
\[
2ab = 2 \cdot 4 \cdot 5 = 40
\]

As we need \(-40x\), which is negative, we must take \(b\) as \(-5\).

3. Confirm the trinomial:

Now, let's express \(16x^2 - 40x + 25\) in terms of \((4x - 5)^2\):

\[
(4x - 5)^2 = 4^2 x^2 - 2 \cdot 4 \cdot 5 x + 5^2 = 16x^2 - 40x + 25
\]

These calculations confirm that the original trinomial can indeed be rewritten as:

\[
16x^2 - 40x + 25 = (4x - 5)^2
\]

Therefore, the factored form of the expression \(16x^2 - 40x + 25\) is:

\[
\boxed{(4x - 5)^2}
\]