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 polynomials. In this article, we will discuss what perfect square trinomials are, how to identify them, and provide examples to illustrate the concept.
What are Perfect Square Trinomials?
A perfect square trinomial is a quadratic expression that can be written in the form of:
(a + b)(a + b) = a^2 + 2ab + b^2
where 'a' and 'b' are constants. This expression can be simplified to:
a^2 + 2ab + b^2
A perfect square trinomial is a special type of quadratic expression that can be factored into the square of a binomial. It is called "perfect" because it can be expressed as the square of a single binomial.
How to Identify Perfect Square Trinomials
To identify a perfect square trinomial, we need to look for the following characteristics:
- The expression must be a quadratic expression, i.e., it must have a squared variable (x^2).
- The expression must have a constant term (b^2).
- The expression must have a middle term (2ab) that is twice the product of the square roots of the constant term.
Examples of Perfect Square Trinomials
Let's consider the following examples:
Example 1: x^2 - 9
This expression can be written as:
(x - 3)(x - 3) = x^2 - 6x + 9
However, this is not a perfect square trinomial because the middle term (-6x) is not twice the product of the square roots of the constant term (9).
Example 2: x^2 - 100
This expression can be written as:
(x - 10)(x - 10) = x^2 - 20x + 100
However, this is not a perfect square trinomial because the middle term (-20x) is not twice the product of the square roots of the constant term (100).
Example 3: x^2 - 4x + 4
This expression can be written as:
(x - 2)(x - 2) = x^2 - 4x + 4
This is a perfect square trinomial because the middle term (-4x) is twice the product of the square roots of the constant term (4).
Example 4: x^2 + 10x + 25
This expression can be written as:
(x + 5)(x + 5) = x^2 + 10x + 25
This is a perfect square trinomial because the middle term (10x) is twice the product of the square roots of the constant term (25).
Example 5: x^2 + 15x + 36
This expression cannot be written as the square of a binomial, so it is not a perfect square trinomial.
Conclusion
In conclusion, a perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It is characterized by the presence of a squared variable, a constant term, and a middle term that is twice the product of the square roots of the constant term. We have provided examples to illustrate the concept and have identified the perfect square trinomials among the given options.
Which are Perfect Square Trinomials?
Based on the discussion above, the perfect square trinomials among the given options are:
- C. x^2 - 4x + 4
- D. x^2 + 10x + 25
Q&A: Perfect Square Trinomials
Q: What is a perfect square trinomial?
A: A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It is a special type of quadratic expression that can be expressed as the square of a single binomial.
Q: How do I identify a perfect square trinomial?
A: To identify a perfect square trinomial, look for the following characteristics:
- The expression must be a quadratic expression, i.e., it must have a squared variable (x^2).
- The expression must have a constant term (b^2).
- The expression must have a middle term (2ab) that is twice the product of the square roots of the constant term.
Q: What are the characteristics of a perfect square trinomial?
A: A perfect square trinomial has the following characteristics:
- It is a quadratic expression.
- It has a constant term (b^2).
- It has a middle term (2ab) that is twice the product of the square roots of the constant term.
Q: How do I factor a perfect square trinomial?
A: To factor a perfect square trinomial, follow these steps:
- Identify the square root of the constant term.
- Write the expression as the square of a binomial, using the square root of the constant term as the variable.
- Simplify the expression to get the factored form.
Q: Can you provide examples of perfect square trinomials?
A: Yes, here are some examples of perfect square trinomials:
- x^2 - 4x + 4: This expression can be factored as (x - 2)(x - 2).
- x^2 + 10x + 25: This expression can be factored as (x + 5)(x + 5).
- x^2 - 9: This expression can be factored as (x - 3)(x - 3).
Q: Can you provide examples of non-perfect square trinomials?
A: Yes, here are some examples of non-perfect square trinomials:
- x^2 - 100: This expression cannot be factored as the square of a binomial.
- x^2 + 15x + 36: This expression cannot be factored as the square of a binomial.
Q: Why are perfect square trinomials important?
A: Perfect square trinomials are important because they can be factored into the square of a binomial, making them easier to solve and manipulate. They are also used in various mathematical applications, such as solving quadratic equations and factoring polynomials.
Q: Can you provide a formula for perfect square trinomials?
A: Yes, the formula for perfect square trinomials is:
(a + b)(a + b) = a^2 + 2ab + b^2
This formula can be used to identify and factor perfect square trinomials.
Conclusion
In conclusion, perfect square trinomials are a special type of quadratic expression that can be factored into the square of a binomial. They have specific characteristics and can be identified using the formula (a + b)(a + b) = a^2 + 2ab + b^2. We hope this Q&A article has provided you with a comprehensive understanding of perfect square trinomials.