Which Expressions Are Perfect Square Trinomials? Check All That Apply.- { X^2 + 16x + 8$}$- { X^2 + 14x + 49$}$- { X^2 - 5x + 25$}$- { X^2 - 24x + 144$}$- { X^2 + 9x - 81$}$

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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 explore the characteristics of perfect square trinomials and identify which of the given expressions fit this category.

What is a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression that can be written in the form of (a+b)2{(a + b)^2} or (a−b)2{(a - b)^2}, where a{a} and b{b} are constants. When expanded, a perfect square trinomial takes the form of a2+2ab+b2{a^2 + 2ab + b^2} or a2−2ab+b2{a^2 - 2ab + b^2}. The key characteristic of a perfect square trinomial is that it has a constant term that is the square of the coefficient of the linear term.

Characteristics of Perfect Square Trinomials

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

  • The expression must be a quadratic expression, meaning it must have a squared variable term.
  • The expression must have a constant term that is the square of the coefficient of the linear term.
  • The expression must have a linear term that is twice the product of the coefficients of the squared variable term and the constant term.

Analyzing the Given Expressions

Now, let's analyze each of the given expressions to determine which ones are perfect square trinomials.

{x^2 + 16x + 8$}$

To determine if this expression is a perfect square trinomial, we need to check if it meets the characteristics mentioned earlier. The expression has a squared variable term, but the constant term is not the square of the coefficient of the linear term. Therefore, this expression is not a perfect square trinomial.

{x^2 + 14x + 49$}$

This expression has a squared variable term and a constant term that is the square of the coefficient of the linear term. However, the linear term is not twice the product of the coefficients of the squared variable term and the constant term. Therefore, this expression is not a perfect square trinomial.

{x^2 - 5x + 25$}$

This expression has a squared variable term and a constant term that is the square of the coefficient of the linear term. The linear term is also twice the product of the coefficients of the squared variable term and the constant term. Therefore, this expression is a perfect square trinomial.

{x^2 - 24x + 144$}$

This expression has a squared variable term and a constant term that is the square of the coefficient of the linear term. However, the linear term is not twice the product of the coefficients of the squared variable term and the constant term. Therefore, this expression is not a perfect square trinomial.

{x^2 + 9x - 81$}$

This expression has a squared variable term and a constant term that is the square of the coefficient of the linear term. However, the linear term is not twice the product of the coefficients of the squared variable term and the constant term. Therefore, this expression is not a perfect square trinomial.

Conclusion

In conclusion, only one of the given expressions is a perfect square trinomial, which is {x^2 - 5x + 25$}$. This expression meets the characteristics of a perfect square trinomial, including having a squared variable term, a constant term that is the square of the coefficient of the linear term, and a linear term that is twice the product of the coefficients of the squared variable term and the constant term.

Perfect Square Trinomials: A Summary

Expression Perfect Square Trinomial
{x^2 + 16x + 8$}$ No
{x^2 + 14x + 49$}$ No
{x^2 - 5x + 25$}$ Yes
{x^2 - 24x + 144$}$ No
{x^2 + 9x - 81$}$ No

Final Thoughts

Introduction

In our previous article, we explored the characteristics of perfect square trinomials and identified which of the given expressions fit this category. In this article, we will provide a comprehensive Q&A guide to help you better understand perfect square trinomials.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a quadratic expression that can be written in the form of (a+b)2{(a + b)^2} or (a−b)2{(a - b)^2}, where a{a} and b{b} are constants. When expanded, a perfect square trinomial takes the form of a2+2ab+b2{a^2 + 2ab + b^2} or a2−2ab+b2{a^2 - 2ab + b^2}.

Q: What are the characteristics of a perfect square trinomial?

A: The characteristics of a perfect square trinomial are:

  • The expression must be a quadratic expression, meaning it must have a squared variable term.
  • The expression must have a constant term that is the square of the coefficient of the linear term.
  • The expression must have a linear term that is twice the product of the coefficients of the squared variable term and the constant term.

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, you need to check if it meets the characteristics mentioned earlier. You can do this by:

  • Checking if the expression has a squared variable term.
  • Checking if the constant term is the square of the coefficient of the linear term.
  • Checking if the linear term is twice the product of the coefficients of the squared variable term and the constant term.

Q: What are some examples of perfect square trinomials?

A: Some examples of perfect square trinomials include:

  • (x+3)2=x2+6x+9{(x + 3)^2 = x^2 + 6x + 9}
  • (x−4)2=x2−8x+16{(x - 4)^2 = x^2 - 8x + 16}
  • (x+2)2=x2+4x+4{(x + 2)^2 = x^2 + 4x + 4}

Q: Can a perfect square trinomial be factored?

A: Yes, a perfect square trinomial can be factored into the square of a binomial. For example:

  • x2+6x+9=(x+3)2{x^2 + 6x + 9 = (x + 3)^2}
  • x2−8x+16=(x−4)2{x^2 - 8x + 16 = (x - 4)^2}
  • x2+4x+4=(x+2)2{x^2 + 4x + 4 = (x + 2)^2}

Q: What are some common mistakes to avoid when identifying perfect square trinomials?

A: Some common mistakes to avoid when identifying perfect square trinomials include:

  • Failing to check if the expression has a squared variable term.
  • Failing to check if the constant term is the square of the coefficient of the linear term.
  • Failing to check if the linear term is twice the product of the coefficients of the squared variable term and the constant term.

Q: How can I practice identifying perfect square trinomials?

A: You can practice identifying perfect square trinomials by:

  • Working through examples and exercises in your textbook or online resources.
  • Creating your own examples and exercises to practice identifying perfect square trinomials.
  • Using online tools or apps to practice identifying perfect square trinomials.

Conclusion

In conclusion, perfect square trinomials are an essential concept in algebra, and identifying them is crucial for solving quadratic equations and factoring expressions. By understanding the characteristics of perfect square trinomials and practicing identifying them, you can become proficient in this area and improve your math skills.

Perfect Square Trinomials: A Summary

Expression Perfect Square Trinomial
{x^2 + 16x + 8$}$ No
{x^2 + 14x + 49$}$ No
{x^2 - 5x + 25$}$ Yes
{x^2 - 24x + 144$}$ No
{x^2 + 9x - 81$}$ No

Final Thoughts

Perfect square trinomials are a fundamental concept in algebra, and identifying them is crucial for solving quadratic equations and factoring expressions. By understanding the characteristics of perfect square trinomials and practicing identifying them, you can become proficient in this area and improve your math skills.