## Perfect Square Trinomial

Perfect square trinomials are algebraic expressions with three terms that are obtained by multiplying a binomial with the same binomial. A perfect square is a number that is obtained by multiplying a number by itself. Binomials are algebraic expressions consisting of just two terms that are either separated by a positive (+) or a negative (-) sign. Similarly, trinomials are algebraic expressions consisting of three terms. When a binomial consisting of a variable and a constant is multiplied by itself, it results in a perfect square trinomial having three terms. The terms of a perfect square trinomial are separated by either a positive or a negative sign.

## Perfect Square Trinomial Definition

A perfect square trinomial is defined as an algebraic expression that is obtained by squaring a binomial expression. It is of the form ax^{2} + bx + c. Here a, b, and c are real numbers and a ≠ 0. For example, let us take a binomial (x+4) and multiply it to (x+4). The result obtained is x^{2} + 8x + 16. A perfect square trinomial can be decomposed into two binomials and the binomials when multiplied with each other gives the perfect square trinomial.

## Perfect Square Trinomial Pattern

Perfect square trinomial generally has the pattern of a^{2} + 2ab + b^{2}or a^{2} – 2ab + b^{2}. Given a binomial, to find the perfect square trinomial, we follow the steps given below. They are,

**Step 1:**Find the square the first term of the binomial.**Step 2:**Multiply the first and the second term of the binomial with 2.**Step 3:**Find the square of the second term of the binomial.**Step 4:**Sum up all the three terms obtained in steps 1, 2, and 3.

The first term of the perfect square trinomial is the square of the first term of the binomial. and the second term is twice the product of the two terms of the binomial and the third term is the square of the second term of the binomial. Take a look at the figure shown below to understand the perfect square trinomial pattern. If the binomial being squared has a positive sign, then all the terms in the perfect square trinomial are positive, whereas, if the binomial has a negative sign attached with its second term, then the second term of the trinomial (which is twice the product of the two variables) will be negative.

## How to Factor Perfect Square Trinomial?

Perfect square trinomials are either separated by a positive or a negative symbol between the terms. Two important algebraic identities with regards to perfect square trinomial are as follows.

- (a + b)
^{2}= a^{2}+ 2ab + b^{2} - (a – b)
^{2}= a^{2}– 2ab + b^{2}

The steps to be followed to factor a perfect square polynomial are as follows.

- Write the given perfect square trinomial of the form a
^{2}+ 2ab + b^{2}or a^{2}– 2ab + b^{2}, such that the first and the third terms are perfect squares, one being a variable and another being a constant. - Check if the middle term is twice the product of the first and the third term. Also, check the sign of the middle term.
- If the middle term is positive, then compare the perfect square trinomial with a
^{2}+ 2ab + b^{2}and if the middle term is negative, then compare the perfect square trinomial with a^{2}– 2ab + b^{2}. - If the middle term is positive, then the factors are (a+b) (a+b) and if the middle term is negative, then the factors are (a-b) (a-b).

## Perfect Square Trinomial Formula

Perfect square trinomial is obtained by multiplying the same binomial expression with each other. A trinomial is said to be a perfect square if it is of the form ax^{2}+bx+c and also satisfies the condition of b^{2} = 4ac. There are two forms of a perfect square trinomial. They are,

- (ax)
^{2}+ 2abx + b^{2}= (ax + b)^{2}—– (1) - (ax)
^{2}−2abx + b^{2}= (ax−b)^{2}—– (2)

For example, let us take a perfect square polynomial, x^{2} + 6x + 9. Comparing this with the form ax^{2}+bx+c, we get a = 1, b = 6 and c = 9. Let us check if this trinomial satisfies the condition b^{2 }= 4ac.

b^{2} = 36 and 4 × a × c = 4 × 1 × 9, which is equal to 36.

Therefore, the trinomial satisfies the condition b^{2} = 4ac. So, we can call it a perfect square trinomial.

## Perfect Square Trinomial Examples

**Example 1: **Factor the perfect square trinomial x^{2} + 10x + 25.

**Solution:**

We can see that the given expression x^{2} + 10x + 25 is of the form a^{2} + 2ab + b^{2}. Factors of (a^{2} + 2ab + b^{2}) are (a+b) (a+b).

Here, a = x and b = 5 and 2ab = 10x.

Therefore, the factors are (x + 5) (x + 5) or (x + 5)^{2}.

**Example 2: **Find the factors of the perfect square trinomial 9x^{2} – 6x + 1.

**Solution:**

We can see that the given equaion is of the form a^{2} – 2ab + b^{2}. Factors of (a^{2} – 2ab + b^{2}) are (a-b) (a-b).

Here, a = 3x, 2ab = 6x and b = 1

Therefore, the factors of the perfect square trinomial are (3x -1) (3x -1) or (3x -1)^{2}.

**Example 3: **Find if 4x^{2} + 8x + 4 is a perfect square trinomial or not.

**Solution:**

To find if a given expression is a perfect square trinomial or not, we check if it satisfies the condition b^{2} = 4ac.

Given equation = 4x^{2} + 8x + 4.

Comparing this equation to ax^{2 }+ bx + c , we get a = 4, b = 8 and c = 4.

Therefore, b^{2} = 8 × 8 (or) 64 —–(1)

4×a×c = 4 × 4 × 4 (or) 64 ——(2)

From (1) and (2), we find that b^{2} = 4ac.

Therefore, 4x^{2} + 8x + 4 is a perfect square trinomial.

## Perfect Square Trinomial FAQs

### What is a Perfect Square Trinomial?

A perfect square trinomial is an algebraic expression that is of the form ax^{2} + bx + c, which has three terms. It is obtained by the multiplication of a binomial with itself. For example, x^{2} + 6x + 9 is a perfect square polynomial obtained by multiplying the binomial (x + 3) by itself. In other words, (x +3) (x + 3) = x^{2} + 6x + 9.

### How is Perfect Square Trinomial Formed?

A perfect square trinomial is formed by multiplying two binomials, which are one and the same. A binomial is an algebraic expression with two terms and a trinomial is an algebraic expression with three terms. For example, if (a + 2) is a binomial, then the perfect square trinomial is obtained by multiplying (a+2) and (a+2), which gives a^{2} + 4a + 4.

### Should All the Three Terms in an Algebraic Expression Be Squares to Call it a Perfect Square Trinomial?

No, only the first and the third terms should be perfect squares for an algebraic expression to be a perfect square trinomial, For example, in the expression x^{2} + 2x + 1, the first term is the square of ‘x’ and the third term is the square of ‘1’.

### What is the Pattern of a Perfect Square Trinomial?

The perfect square trinomial is either of the pattern a^{2} + 2ab + b^{2} or a^{2} – 2ab + b^{2 }. These expressions are obtained by squaring the binomials (a+b) and (a-b) respectively.

### What are the Important Algebraic Expressions to be Known to Factor a Perfect Square Trinomial?

The important algebraic expressions to be known for factoring a perfect square trinomial are,

- (a+b)
^{2}= a^{2}+ 2ab + b^{2} - (a-b)
^{2}= a^{2}−2ab + b^{2}

### What is the Formula for Perfect Square Trinomial?

A perfect square trinomial is obtained by multiplying two same binomials. It takes the form of the following two expressions. They are,

- (ax)
^{2}+ 2abx + b^{2}= (ax + b)^{2} - (ax)
^{2}−2abx + b^{2}= (ax−b)^{2}

Also, an expression is said to be a perfect square trinomial if it is of the form ax^{2}+bx+c and b^{2} = 4ac is satisfied. For example, comparing the expression x^{2} + 2x + 1, to the forms mentioned above, we find that a = 1, b = 2 and c = 1, therefore, b^{2} = 4 and 4 × a × c = 4 × 1 × 1, which is 4. Therefore, b^{2} = 4ac is satisfied and we conclude that x^{2} + 2x + 1 is a perfect square trinomial.

### How do you Factor a Perfect Square Trinomial?

A perfect square trinomial has three terms which may be in the form of (ax)^{2}+ 2abx + b^{2}= (ax + b)^{2} (or) (ax)^{2}−2abx + b^{2} = (ax−b)^{2}.The steps to be followed to factor a perfect square polynomial are as follows.

- Verify whether the given perfect square trinomial is of the form a
^{2}+ 2ab + b^{2}(or) a^{2}−2ab + b^{2}. - Check if the middle term is twice the product of the first and the third term. Also, check the sign of the middle term.
- If the middle term is positive, then compare the perfect square trinomial with a
^{2}+ 2ab + b^{2}and if the middle term is negative, then compare the perfect square trinomial with a^{2}– 2ab + b^{2}. - If the middle term is positive, then the factors are (a+b) (a+b) and if the middle term is negative, then the factors are (a-b) (a-b).

### Can Perfect Square Trinomials be called Quadratic Equations?

Yes, perfect square polynomials can be of the form of quadratic equations. A quadratic equation consists of one squared term and it should have the degree of 2. Since perfect square trinomials have their degree as 2, we can call them quadratic equations. Note that all quadratic equations cannot be considered as perfect square trinomials as all quadratic equations do not satisfy the conditions required for a perfect square trinomial.

### Can All Algebraic Expressions that have the First and the Last Terms as Perfect Squares be called Perfect Square Trinomials?

No, not all the algebraic expressions that have the first and the last term as perfect squares be called perfect square trinomials. Let us understand this with an example, x^{2} + 18x + 36 is an algebraic expression with three terms and the first and third terms are perfect squares. In that case, comparing this expression to a^{2} + 2ab + b^{2 } we get a = 1 and b = 6, so 2ab should be 12x for this expression to be a perfect square trinomial. But here the value of 2ab = 18x, therefore, we cannot say that all algebraic expressions with first and third terms as squares are perfect square trinomials.

### Factoring Perfect Square Trinomials

### What is a perfect square trinomial

### Factoring Perfect Square Trinomials – Ex1

### Perfect Square Trinomial – Explanation & Examples

A quadratic equation is a second degree polynomial usually in the form of f(x) = ax^{2} + bx + c where a, b, c, ∈ R, and a ≠ 0. The term ‘a’ is referred to as the leading coefficient, while ‘c’ is the absolute term of f (x).

Every quadratic equation has two values of the unknown variable, usually known as the roots of the equation (α, β). We can obtain the roots of a quadratic equation by factoring the equation.

### What is a Perfect Square Trinomial?

The ability to **recognize special cases of polynomials** that we can easily factor in is a fundamental skill for solving any algebraic expressions that involve polynomials.

One of these “**easy to factor**” polynomials is the perfect square trinomial. We can recall that a trinomial is an algebraic expression composed of three terms connected by addition or subtraction.

Similarly, a binomial is an expression **composed of two terms**. Therefore, a perfect square trinomial can be defined as an expression that is obtained by squaring a binomial

Learning** how to recognize a perfect square trinomial** is the first step to factoring it.

*The following are the tips on how to recognize a perfect square trinomial:*

- Check whether the first and last terms of the trinomial are perfect squares.
- Multiply the roots of the first and third terms together.
- Compare to the middle terms with the result in step two
- If the first and last terms are perfect squares, and the middle term’s coefficient is twice the product of the square roots of the first and last terms, then the expression is a perfect square trinomial.

### How to Factor a Perfect Square Trinomial?

Once you have identified a perfect square trinomial, factoring it is quite a straightforward process.

*Let’s take a look at the steps for factoring a perfect square trinomial.*

- Identify the squared numbers in the first and third terms of the trinomial.
- Examine the middle term if it has either positive or negative. If the middle term of the trinomial is positive or negative, then the factors will have a plus and minus sign, respectively.
- Write out your terms by applying the following identities:

(i) a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b)

(ii) a^{2} – 2ab + b^{2} = (a – b)^{2} = (a – b) (a – b)

### Perfect Square Trinomial Formula

An expression obtained from the square of a binomial equation is a perfect square trinomial. An expression is said to a perfect square trinomial if it takes the form ax^{2} + bx + c and satisfies the condition b^{2} = 4ac.

*The perfect square formula takes the following forms:*

- (ax)
^{2 }+ 2abx + b^{2 }= (ax + b)^{2} - (ax)
^{2 }−2abx + b^{2}= (ax−b)^{2}

*Example 1*

Factor x^{2}+ 6x + 9

Solution

We can rewrite the expression x^{2} + 6x + 9 in the form a^{2} + 2ab + b^{2 }as;

x^{2}+ 6x + 9 ⟹ (x)^{2} + 2 (x) (3) + (3)^{2}

Applying the formula of a^{2} + 2ab + b^{2} = (a + b)^{2} to the expression gives;

= (x + 3)^{2}

= (x + 3) (x + 3)

*Example 2*

Factor x^{2} + 8x + 16

Solution

Write the expression x^{2} + 8x + 16 as a^{2} + 2ab + b^{2}

x^{2} + 8x + 16 ⟹ (x)^{2} + 2 (x) (4) + (4)^{2}

Now we will apply the perfect square trinomial formula;

= (x + 4)^{2}

= (x + 4) (x + 4)

*Example 3*

Factor 4a^{2} – 4ab + b^{2}

Solution

4a^{2} – 4ab + b^{2} ⟹ (2a)^{2} – (2)(2) ab + b^{2}

= (2a – b)^{2}

= (2a – b) (2a – b)

*Example 4*

Factor 1- 2xy- (x^{2} + y^{2})

Solution

1- 2xy- (x^{2} + y^{2})

= 1 – 2xy – x^{2} – y^{2}

= 1 – (x^{2} + 2xy + y^{2})

= 1 – (x + y )^{2}

= (1)^{2} – (x + y)^{2}

= [1 + (x + y)] [1 – (x + y)]

= [1 + x + y] [1 – x – y]

*Example 5*

Factor 25y^{2} – 10y + 1

Solution

25y^{2} – 10y + 1⟹ (5y)^{2} – (2)(5)(y)(1) + 1^{2}

= (5y – 1)^{2}

= (5y– 1) (5y – 1)

*Example 6*

Factor 25t^{2} + 5t/2 + 1/16.

Solution

25t^{2} + 5t/2 + 1/16 ⟹ (5t)^{2 }+ (2)(5)(t) (1/4) + (1/4)^{2}

= (5t + 1/4)^{2}

= (5t + 1/4) (5t + 1/4)

*Example 7*

Factor x^{4} – 10x^{2}y^{2} + 25y^{4}

Solution

x^{4} – 10x^{2}y^{2} + 25y^{4} ⟹ (x^{2})^{2} – 2 (x^{2}) (5y^{2}) + (5y^{2})^{2}

Apply the formula a^{2} + 2ab + b^{2} = (a + b)^{2} to get,

= (x^{2} – 5y^{2})^{2}

= (x^{2} – 5y^{2}) (x^{2} – 5y^{2})

### Solved Example

**Question: **Is the trinomial x^{2 }– 6x + 9 a perfect square?

**Solution:**

x^{2} – 6x + 9

= x^{2} – 3x – 3x + 9

= x(x – 3) – 3(x – 3)

= (x – 3)(x – 3)

Alternatively,

x^{2} – 6x + 9 = x^{2} – 2(3)(x) + 3^{2}

= (x – 3)^{2}

The factors of the given equation are a perfect square.

So, the given trinomial is a perfect square.

### The Square of a Binomial

With perfect square trinomials, you will need to be able to move forwards and backwards. You should be able to take the binomials and find the perfect square trinomial and you should be able to take the perfect square trinomials and create the binomials from which it came. Any time you take a binomial and multiply it to itself, you end up with a perfect square trinomial. For example, take the binomial (*x* + 2) and multiply it by itself (*x* + 2).

(*x* + 2)(*x* + 2) = *x*^{2} + 4*x* + 4

The result is a perfect square trinomial.

To find the perfect square trinomial from the binomial, you will follow four steps:

Step One: Square the *a*

Step Two: Square the *b*

Step Three: Multiply 2 by *a* by *b*

Step Four: Add *a*^{2}, *b*^{2}, and 2*ab*

(*a* + *b*)^{2} = *a*^{2} + 2*ab* + *b*^{2}

Let’s add some numbers now and find the perfect square trinomial for 2*x* – 3*y*. For this:

*a* = 2*x**b* = 3*y*

Step One: Square the *a**a*^{2} = 4*x*^{2}

Step Two: Square the *b**b*^{2} = 9*y*^{2}

Step Three: Multiply 2 by *a* by *‘b*

2(2*x*)(-3*y*) = -12*xy*

Step Four: Add *a*^{2}, *b*^{2}, and 2*ab*

4*x*^{2} – 12*xy* + 9*y*^{2}

### How do you square a trinomial?

Squaring a **Trinomial**

To **square** a **trinomial**, all we have to do is follow these two steps: Identify a as the first term in the **trinomial**, b as the second term, and c as the third term. Plug a, b, and c into the formula.

### What is perfect square example?

In mathematics, a **square** number or **perfect square** is an integer that is the **square** of an integer; in other words, it is the product of some integer with itself. For **example**, 9 is a **square** number, since it can be written as 3 × 3.

### What is the square of a binomial?

Definition of a Perfect **Square Binomial**

A perfect **square binomial** is a trinomial that when factored gives you the **square of a binomial**. For example, the trinomial x^2 + 2xy + y^2 is a perfect **square binomial** because it factors to (x + y)^2. … Also, look at the first and last term of the trinomial.

### Perfect Square Trinomial Calculator

We have already discussed perfect square trinomials: |

### What Is A Perfect Square Trinomial

An expression obtained from the square of binomial equation is a perfect square trinomial. If a trinomial is in the form ax^{2} + bx + cis said to be perfect square, if only it satisfies the condition b^{2} = 4ac.

The Perfect Square Trinomial Formula is given as, (ax)2+2abx+b2=(ax+b)2 (ax)2−2abx+b2=(ax−b)2

**Question: **Is the x^{2 }– 6x + 9 a perfect square?

**Solution:**

x^{2} – 6x + 9

= x^{2} – 3x – 3x + 9

= x(x – 3) – 3(x – 3)

= (x – 3)(x – 3)

The factors of the given equation are a perfect square.

So, the is a perfect square.

### Perfect Square Trinomial Formula

With perfect square trinomials, you will need to be able to move forwards and backward. You should be able to take the binomials and find the perfect square and you should be able to make the perfect square and create the binomials from which it came. Any time you take a binomial and multiply it to itself, you end up with a perfect square. For example, take the binomial (*x* + 2) and multiply it by itself (*x* + 2).

(*x* + 2)(*x* + 2) = *x*2 + 4*x* + 4

The result is a perfect square.

To find the perfect square from the binomial, you will follow four steps:

Step One: Square the *a*

Step Two: Square the *b*

Step Three: Multiply 2 by *a* by *b*

Step Four: Add *a*2, *b*2, and 2*ab*

(*a* + *b*)2 = *a*2 + 2*ab* + *b*2

Let’s add some numbers now and find the perfect square for 2*x* – 3*y*. For this:

*a* = 2*x**b* = 3*y*

Step One: Square the *a**a*2 = 4*x*2

Step Two: Square the *b**b*2 = 9*y*2

Step Three: Multiply 2 by *a* by *‘b*

2(2*x*)(-3*y*) = -12*xy*

Step Four: Add *a*2, *b*2, and 2*ab*

4*x*2 – 12*xy* + 9*y*2

### Perfect Square Trinomial Definition

Before we can get to defining a perfect square, we need to review some vocabulary.

Perfect squares are numbers or expressions that are the product of a number or expression multiplied to itself. 7 times 7 is 49, so 49 is a perfect square. *x* squared times *x* squared equals *x* to the fourth, so *x* to the fourth is a perfect square.

- Binomials are algebraic expressions containing only two terms. Example:
*x*+ 3 - Trinomials are algebraic expressions that contain three terms. Example: 3
*x*2 + 5*x*– 6

**Perfect square trinomials** are algebraic expressions with three terms that are created by multiplying a binomial to itself. Example: (3*x* + 2*y*)2 = 9*x*2 + 12*xy* + 4*y*2

Recognizing when you have these perfect square trinomials will make factoring them much simpler. They are also very helpful when solving and graphing certain kinds of equations.

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