Which Shows A Perfect Square Trinomial?A. $50y^2 - 4x^2$B. $100 - 36x^2y^2$C. $16x^2 + 24xy + 9y^2$D. $49x^2 - 70xy + 10y^2$

Understanding Perfect Square Trinomials

A perfect square trinomial is a polynomial expression that can be factored into the square of a binomial. It is a trinomial that can be written in the form of $(a + b)^2$ or $(a - b)^2$, where $a$ and $b$ are expressions. In this article, we will explore which of the given options shows a perfect square trinomial.

What is a Perfect Square Trinomial?

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

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

or

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

where $a$ and $b$ are expressions.

How to Identify a Perfect Square Trinomial

To identify a perfect square trinomial, we need to check if it can be factored into the square of a binomial. We can do this by looking for the following patterns:

• 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.

If a trinomial satisfies these conditions, it is a perfect square trinomial.

Analyzing the Options

Let's analyze each of the given options to determine which one shows a perfect square trinomial.

Option A: $50y^2 - 4x^2$

This option does not show a perfect square trinomial because it does not satisfy the conditions mentioned earlier. The first and last terms are not perfect squares, and the middle term is not twice the product of the square roots of the first and last terms.

Option B: $100 - 36x^2y^2$

This option does not show a perfect square trinomial because it does not satisfy the conditions mentioned earlier. The first and last terms are not perfect squares, and the middle term is not twice the product of the square roots of the first and last terms.

Option C: $16x^2 + 24xy + 9y^2$

This option shows a perfect square trinomial because it satisfies the conditions mentioned earlier. 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.

Option D: $49x^2 - 70xy + 10y^2$

This option does not show a perfect square trinomial because it does not satisfy the conditions mentioned earlier. The first and last terms are not perfect squares, and the middle term is not twice the product of the square roots of the first and last terms.

Conclusion

In conclusion, the option that shows a perfect square trinomial is Option C: $16x^2 + 24xy + 9y^2$. This option satisfies the conditions mentioned earlier, and it can be factored into the square of a binomial.

Perfect Square Trinomial Formula

The perfect square trinomial formula is:

$(a + b)^2 = a^2 + 2ab + b^2$

or

$(a - b)^2 = a^2 - 2ab + b^2$

where $a$ and $b$ are expressions.

Example of Perfect Square Trinomial

Let's consider an example of a perfect square trinomial:

$(3x + 2y)^2 = 9x^2 + 12xy + 4y^2$

This expression is a perfect square trinomial because it satisfies the conditions mentioned earlier. 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.

Real-World Applications of Perfect Square Trinomials

Perfect square trinomials have many real-world applications in mathematics and science. They are used to solve quadratic equations, find the area and perimeter of shapes, and model real-world phenomena.

Solving Quadratic Equations

Perfect square trinomials can be used to solve quadratic equations. For example, consider the quadratic equation:

$x^2 + 6x + 9 = 0$

This equation can be factored into a perfect square trinomial:

$(x + 3)^2 = 0$

Solving for $x$, we get:

$x = -3$

Finding the Area and Perimeter of Shapes

Perfect square trinomials can be used to find the area and perimeter of shapes. For example, consider a square with side length $x$. The area of the square is given by:

$x^2 + 2x^2 + x^2 = 4x^2$

This expression is a perfect square trinomial. The area of the square is $4x^2$.

Modeling Real-World Phenomena

Perfect square trinomials can be used to model real-world phenomena. For example, consider a projectile motion problem. The height of the projectile is given by:

$h(t) = 16t^2 + 32t + 16$

This expression is a perfect square trinomial. The height of the projectile is given by $h(t) = 16(t + 2)^2$.

Conclusion

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about perfect square trinomials.

Q: What is a perfect square trinomial?

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

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, you need to check if it can be factored into the square of a binomial. You can do this by looking for the following patterns:

• 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: What are some examples of perfect square trinomials?

A: Some examples of perfect square trinomials include:

• $(x + 3)^2 = x^2 + 6x + 9$
• $(x - 2)^2 = x^2 - 4x + 4$
• $(2x + 1)^2 = 4x^2 + 4x + 1$

Q: Can I use perfect square trinomials to solve quadratic equations?

A: Yes, you can use perfect square trinomials to solve quadratic equations. For example, consider the quadratic equation:

$x^2 + 6x + 9 = 0$

This equation can be factored into a perfect square trinomial:

$(x + 3)^2 = 0$

Solving for $x$, we get:

$x = -3$

Q: Can I use perfect square trinomials to find the area and perimeter of shapes?

A: Yes, you can use perfect square trinomials to find the area and perimeter of shapes. For example, consider a square with side length $x$. The area of the square is given by:

$x^2 + 2x^2 + x^2 = 4x^2$

This expression is a perfect square trinomial. The area of the square is $4x^2$.

Q: Can I use perfect square trinomials to model real-world phenomena?

A: Yes, you can use perfect square trinomials to model real-world phenomena. For example, consider a projectile motion problem. The height of the projectile is given by:

$h(t) = 16t^2 + 32t + 16$

This expression is a perfect square trinomial. The height of the projectile is given by $h(t) = 16(t + 2)^2$.

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

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

• Not checking if the first and last terms are perfect squares.
• Not checking if the middle term is twice the product of the square roots of the first and last terms.
• Not factoring the trinomial correctly.

Q: How can I practice working with perfect square trinomials?

A: You can practice working with perfect square trinomials by:

• Factoring perfect square trinomials.
• Solving quadratic equations using perfect square trinomials.
• Finding the area and perimeter of shapes using perfect square trinomials.
• Modeling real-world phenomena using perfect square trinomials.

Conclusion

In conclusion, perfect square trinomials are an important concept in mathematics and science. They have many real-world applications, including solving quadratic equations, finding the area and perimeter of shapes, and modeling real-world phenomena. By understanding perfect square trinomials, you can solve problems more efficiently and effectively.

Perfect Square Trinomial Practice Problems

Here are some practice problems to help you practice working with perfect square trinomials:

1. Factor the following perfect square trinomial:

$x^2 + 6x + 9$

2. Solve the following quadratic equation using a perfect square trinomial:

$x^2 + 4x + 4 = 0$

3. Find the area of the following square using a perfect square trinomial:

$x^2 + 2x^2 + x^2$

4. Model the following real-world phenomenon using a perfect square trinomial:

$h(t) = 16t^2 + 32t + 16$

Answer Key

1. $(x + 3)^2$
2. $x = -2$
3. $4x^2$
4. $h(t) = 16(t + 2)^2$