**Introduction**

Factoring perfect square trinomials is an important skill to develop when studying algebra. This article will comprehensively overview the topic, including grade appropriateness, the math domain it belongs to, Common Core Standards, key concepts, examples, real-life applications, practice tests, and frequently asked questions.

**Grade Appropriateness**

Factoring perfect square trinomials is typically taught in middle or early high school, falling within the 8th to the 10th-grade range.

**Math Domain**

Factoring perfect square trinomials falls under the domain of Algebra, specifically within the area of quadratic expressions and equations.

**Applicable Common Core Standards**

*CCSS.MATH.CONTENT.HSA.SSE.A.2:* Use the structure of an expression to identify ways to rewrite it.

*CCSS.MATH.CONTENT.HSA.REI.B.4:* Solve quadratic equations in one variable.

**Definition of the Topic**

A perfect square trinomial is a quadratic expression of the form ax^{2} + bx + c, where a and c are perfect squares, and b is twice or double the product of the square roots of a and c. Factoring perfect square trinomials involves expressing them as the square of a binomial.

**Key Concepts**

** Perfect square trinomials** can always be factored into the square of a binomial.

The ** coefficients** of the quadratic and constant terms must be perfect squares.

The ** middle term** is two times the product of the square roots of the quadratic and constant terms.

**Discussion with Illustrative Examples**

Perfect square trinomials are three-term algebraic expressions obtained by multiplying one binomial by the same binomial. Generally, perfect square trinomials are in the form a^{2}+2ab+b^{2} or a^{2} -2ab + b^{2}.

The following conditions must be met to factor perfect square trinomials:

a. Check if the quadratic and constant terms are perfect squares and

b. The middle term is twice/double the product of the square roots of the quadratic and constant terms.

Consider the perfect square trinomial 9x^{2} + 12x + 4.

9x^{2} (the quadratic term) and 4 (the constant term) are perfect squares.

The middle term, 12x, equals twice the product of the square roots of 9 and 4, which are 3x and 2, respectively. Therefore, this trinomial can be factored into the square of a binomial as follows: 9x^{2} + 12x + 4 = (3x + 2)^{2}.

**Examples with Solutions**

**Example 1**

Factor the perfect square trinomial: 16x^{2} – 40x + 25.

**Solution**

Here, $\sqrt{16x^2}$ = 4x while $\sqrt{25}$ = 5. The middle term, 40x, is twice the product of 4x and 5. Taking the sign of the trinomial’s middle term, we’ll have (4x-5)^{2}

16x^{2} – 40x + 25 = (4x – 5)^{2}

**Example 2**

Factor the perfect square trinomial: x^{2} + 14x + 49.

**Solution **

The square root of the first and last term of the trinomial is x and 7, respectively. The middle term, 14x, is twice the product of x and 7. Therefore,

x^{2} + 14x + 49 = (x + 7)^{2}

**Real-life Application with Solution**

A square garden has an area of 9x^{2}+ 30x + 25 square meters. Determine the square garden’s length on each side.

**Solution**

Factoring the area expression, we get:

9x^{2} + 30x + 25 = 3x + 52

Since this is a square, both sides are equal in length. Each side of the square garden is 3x + 5 meters long.

**Practice Test**

1.Factor the perfect square trinomial: 4x^{2}– 20x+25

2. Factor the perfect square trinomial: 49m^{2}+28m+4

3. Factor the perfect square trinomial: 25p^{2}-30p+9

4. Factor the perfect square trinomial: 64y^{2}-32y+ 4

5. Factor the perfect square trinomial: z^{2}+ 18z +81

6. Factor: (x+y)^{2} + 14x + y + 49

7. Calculate the length of each side of the square with an area of 25x^{2} – 90x + 81.

*Answers:*

1. 4x^{2} – 20x + 25 = (2x-5)^{2}

2. 49m^{2} + 28m + 4 = (7m+2)^{2}

3. 25p^{2} – 30p + 9 = (5p-3)^{2}

4. 64y^{2} – 32y + 4 = (8y-2)^{2}

5. x^{2} + 18x + 81 = (x+9)^{2}

6. (x+y)^{2} + 14(x+y) + 49 = ((x+y)+7)^{2} or (x+y+7)^{2}

7. 25x^{2} – 90x + 81 = (5x^{2} – 9)^{2}. Each side of the square is 5x^{2}-9 units long.

**Frequently Asked Questions (FAQs)**

**What is the difference between a perfect square trinomial and a regular trinomial?**

A perfect square trinomial is a special case of a trinomial that can be factored into the square of a binomial. Regular trinomials may or may not be factorable, and if they are, their factors may not be binomial squares.

**Can all trinomials be factored into perfect square trinomials?**

No, not all trinomials can be factored into perfect square trinomials. Only trinomials that meet the criteria of perfect square trinomials (as described in the key concepts section) can be factored in this way.

**How can I identify if a trinomial is a perfect square trinomial?**

Check if the coefficients of the quadratic and constant terms are perfect squares and if the middle term is twice or double the product of the square roots of the quadratic and constant terms. If these conditions are met, the trinomial is a perfect square.

**Can a perfect square trinomial have negative terms?**

Yes, a perfect square trinomial can have negative terms, but the quadratic and constant terms must still be perfect squares. In addition, the middle term must still be twice or double the product of the square roots of the quadratic and constant terms.

**How does factoring perfect square trinomials help in solving quadratic equations?**

Factoring perfect square trinomials helps simplify quadratic equations, making it easier to find the solutions. By expressing a perfect square trinomial as the square of a binomial, you can apply the square root property to isolate the variable and solve for its possible values.

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