Which Are Perfect Square Trinomials? Select Two Options.A. X 2 − 9 X^2 - 9 X 2 − 9 B. X 2 − 100 X^2 - 100 X 2 − 100 C. X 2 − 4 X + 4 X^2 - 4x + 4 X 2 − 4 X + 4 D. X 2 + 10 X + 25 X^2 + 10x + 25 X 2 + 10 X + 25 E. X 2 + 15 X + 36 X^2 + 15x + 36 X 2 + 15 X + 36

Introduction

In algebra, a perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It is a crucial concept in mathematics, particularly in solving quadratic equations and factoring expressions. In this article, we will discuss what perfect square trinomials are, how to identify them, and provide examples of perfect square trinomials.

What are Perfect Square Trinomials?

A perfect square trinomial is a quadratic expression that can be written in the form of $(a + b)^2$ or $(a - b)^2$, where $a$ and $b$ are constants. It is a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is:

$a^2 + 2ab + b^2$

or

$a^2 - 2ab + b^2$

How to Identify Perfect Square Trinomials

To identify a perfect square trinomial, we need to look for the following characteristics:

• The first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

Examples of Perfect Square Trinomials

Let's consider some examples of perfect square trinomials:

Example 1: $x^2 - 9$

$x^2 - 9$ can be written as $(x - 3)^2$, which is a perfect square trinomial.

Example 2: $x^2 - 100$

$x^2 - 100$ can be written as $(x - 10)^2$, which is a perfect square trinomial.

Example 3: $x^2 - 4x + 4$

$x^2 - 4x + 4$ can be written as $(x - 2)^2$, which is a perfect square trinomial.

Example 4: $x^2 + 10x + 25$

$x^2 + 10x + 25$ can be written as $(x + 5)^2$, which is a perfect square trinomial.

Example 5: $x^2 + 15x + 36$

$x^2 + 15x + 36$ cannot be written as a perfect square trinomial.

Conclusion

In conclusion, perfect square trinomials are quadratic expressions that can be factored into the square of a binomial. They have the characteristics of having the first and last terms as perfect squares and the middle term as twice the product of the square roots of the first and last terms. We have discussed what perfect square trinomials are, how to identify them, and provided examples of perfect square trinomials.

Perfect Square Trinomials: A Summary

Option	Perfect Square Trinomial
A	$x^2 - 9$
B	$x^2 - 100$
C	$x^2 - 4x + 4$
D	$x^2 + 10x + 25$
E	$x^2 + 15x + 36$

Based on the discussion above, the perfect square trinomials are:

• $x^2 - 9$
• $x^2 - 100$
• $x^2 - 4x + 4$
• $x^2 + 10x + 25$

Introduction

In our previous article, we discussed what perfect square trinomials are, how to identify them, and provided examples of perfect square trinomials. In this article, we will answer some frequently asked questions about perfect square trinomials.

Q&A

Q: What is the difference between a perfect square trinomial and a quadratic expression?

A: A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, whereas a quadratic expression is a polynomial of degree two that can be written in the form of $ax^2 + bx + c$.

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, look for the following characteristics:

• The first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

Q: Can a perfect square trinomial have a negative middle term?

A: Yes, a perfect square trinomial can have a negative middle term. For example, $x^2 - 4x + 4$ is a perfect square trinomial with a negative middle term.

Q: Can a perfect square trinomial have a zero middle term?

A: Yes, a perfect square trinomial can have a zero middle term. For example, $x^2 + 0x + 4$ is a perfect square trinomial with a zero middle term.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, look for the square root of the first and last terms and write the middle term as twice the product of these square roots. For example, to factor $x^2 - 9$, we can write it as $(x - 3)^2$.

Q: Can a perfect square trinomial be written in the form of $(a + b)^2$ or $(a - b)^2$?

A: Yes, a perfect square trinomial can be written in the form of $(a + b)^2$ or $(a - b)^2$, where $a$ and $b$ are constants.

Q: How do I determine if a quadratic expression is a perfect square trinomial?

A: To determine if a quadratic expression is a perfect square trinomial, look for the characteristics mentioned above. If the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms, then the quadratic expression is a perfect square trinomial.

Common Mistakes to Avoid

When working with perfect square trinomials, there are some common mistakes to avoid:

• Not recognizing the perfect square trinomial form.
• Not factoring the perfect square trinomial correctly.
• Not identifying the square roots of the first and last terms.

Conclusion

In conclusion, perfect square trinomials are an important concept in algebra, and understanding how to identify and factor them is crucial. By following the characteristics and examples provided in this article, you can become proficient in working with perfect square trinomials.

Perfect Square Trinomials: A Summary

Question	Answer
What is the difference between a perfect square trinomial and a quadratic expression?	A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial.
How do I identify a perfect square trinomial?	Look for the first and last terms to be perfect squares and the middle term to be twice the product of the square roots of the first and last terms.
Can a perfect square trinomial have a negative middle term?	Yes.
Can a perfect square trinomial have a zero middle term?	Yes.
How do I factor a perfect square trinomial?	Look for the square root of the first and last terms and write the middle term as twice the product of these square roots.
Can a perfect square trinomial be written in the form of $(a + b)^2$ or $(a - b)^2$?	Yes.
How do I determine if a quadratic expression is a perfect square trinomial?	Look for the characteristics mentioned above.

By following the answers to these questions, you can become proficient in working with perfect square trinomials.