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

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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.

Understanding the Context

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.

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

Key Insights

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 $

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Final Thoughts

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 $: