Question: Factor the expression: $ 16x^2 - 40x + 25 $.

February 22, 2026

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Topic: Factor $ 16x^2 - 40x + 25 $ Keywords: factor quadratic expression, factor $ 16x^2 - 40x + 25 $, complete factoring guide, factor trinomial, algebraic factoring techniques Meta Description: Learn how to factor $ 16x^2 - 40x + 25 $ step-by-step using proven algebraic methods. Discover if it’s a perfect square trinomial and how to write it in factored form.

Factor the Expression $ 16x^2 - 40x + 25 $: A Complete Guide

If you’ve ever stumbled upon the quadratic expression $ 16x^2 - 40x + 25 $, you might wonder: Is this factorable? The good news is that this trinomial is a perfect square trinomial, and yes, it can be factored neatly into a squared binomial.

In this article, we’ll walk through the process of factoring $ 16x^2 - 40x + 25 $ step-by-step and explain why it works. Whether you're a high school student mastering algebra or a learner brushing up on quadratics, this guide will help you understand factoring quadratics efficiently.

Step 1: Recognize the Perfect Square Trinomial Pattern

The expression $ 16x^2 - 40x + 25 $ resembles the general form of a perfect square trinomial:

$$ a^2x^2 - 2abx + b^2 = (ax - b)^2 $$

Let’s identify $ a $ and $ b $ by examining the first and last terms:

First term: $ 16x^2 = (4x)^2 $ → So, $ a = 4 $
Last term: $ 25 = 5^2 $ → So, $ b = 5 $

Now check the middle term:

$$ -2abx = -2(4)(5)x = -40x $$

This matches exactly with the middle term in our expression. Therefore, the trinomial fits the perfect square pattern.

Step 2: Apply the Factoring Formula

Since $ 16x^2 - 40x + 25 = (4x)^2 - 2(4x)(5) + 5^2 $, it factors as:

$$ 16x^2 - 40x + 25 = (4x - 5)^2 $$

Step 3: Verify the Factorization

To ensure correctness, expand $ (4x - 5)^2 $:

$$ (4x - 5)^2 = (4x)^2 - 2(4x)(5) + 5^2 = 16x^2 - 40x + 25 $$

The result matches the original expression, confirming our factorization is correct.

Why This is a Perfect Square Trinomial

Several key hints told us this was a perfect square:

The first and last terms are perfect squares: $16x^2 = (4x)^2$, $25 = 5^2$.
The middle term equals $-2$ times the square root of the first term times the square root of the last term: $-2(4x)(5)$.
The sign of the middle term is negative, indicating a minus sign in the binomial.

Comparison With Other Factoring Methods

While factoring quadratics often involves finding two binomials of the form $ (ax + b)(cx + d) $, perfect square trinomials are special cases that allow shortcut factoring using the square identity. Using this identity saves time and reduces errors.

For expressions like $ ax^2 + bx + c $ that don’t fit perfect squares, methods like factoring by grouping or the quadratic formula apply—but our case is simple and elegant.

Final Factored Form

$$ oxed{16x^2 - 40x + 25 = (4x - 5)^2} $$

Summary

Factor $ 16x^2 - 40x + 25 $ by recognizing it as a perfect square trinomial.
Identify $ a = 4 $, $ b = 5 $, confirming $ (4x - 5)^2 $.
Verify by expanding $ (4x - 5)^2 $ to match the original.
This method is efficient and reliable for identifying factorable perfect squares.

Need more help with factoring? Check out our full guide on factoring quadratics and master algebra with our step-by-step tutorials and video explanations.

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