
Which of the following is a perfect square trinomial?

(A) 5x2 - 5x + 1

(B) 5x2 - 10x + 5

(C) 10x2 - 5x + 5

(D) 25x2 - 10x + 1

(E) 25x2 - 25x + 5

A perfect square trinomial is of the form:

\[
(ax)^2 + 2abx + b^2 = (ax + b)^2
\]

Let's evaluate each option to see if it fits this form.

**(A) \(5x^2 - 5x + 1\):**

1. The first term is \(5x^2\), but this is not a perfect square (5 is not a perfect square).
2. It cannot be written in the form \((ax + b)^2\).

**(B) \(5x^2 - 10x + 5\):**

1. The first term is \(5x^2\), which is not a perfect square.
2. It cannot be written in the form \((ax + b)^2\).

**(C) \(10x^2 - 5x + 5\):**

1. The first term is \(10x^2\), which is not a perfect square.
2. It cannot be written in the form \((ax + b)^2\).

**(D) \(25x^2 - 10x + 1\):**

1. The first term is \(25x^2 = (5x)^2\), which is a perfect square.
2. The last term is \(1 = 1^2\), which is a perfect square.
3. The middle term is \(-10x\), which can be written as \(2 \cdot 5x \cdot (-1)\).

Thus, this can be written as:

\[
(5x - 1)^2 = 25x^2 - 10x + 1
\]

This is a perfect square trinomial.

**(E) \(25x^2 - 25x + 5\):**

1. The first term is \(25x^2\), which is a perfect square.
2. The last term is 5, which is not a perfect square.

So, option (D) is the only perfect square trinomial.

The answer is:

**(D) \(25x^2 - 10x + 1\)**

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the properties of a perfect square trinomial?
2. How do you factor a perfect square trinomial?
3. Can a trinomial be a perfect square if its coefficients are not integers?
4. What are the differences between perfect square trinomials and general trinomials?
5. How can you verify if a given trinomial is a perfect square?

**Tip:** To quickly identify a perfect square trinomial, check if both the first and last terms are perfect squares, and if the middle term equals twice the product of their square roots.

Which of the following is a perfect square trinomial?
(A) 5x^2 - 5x + 1
(B) 5x^2 - 10x + 5
(C) 10x^2 - 5x + 5
(D) 25x^2 - 10x + 1
(E) 25x^2 - 25x + 5

Learn how to identify and solve perfect square trinomials with this detailed solution. This problem covers key algebraic concepts, including recognizing patterns in perfect square trinomials and applying the formula (ax + b)^2. Suitable for students in Grades 9-11.

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