Perfect Square Trinomial Formula

Perfect Square Trinomial: There is one “special” factoring type that can actually be done using the usual methods for factoring, but, for whatever reason, many texts and instructors make a big deal of treating this case separately. “Perfect square trinomials” are quadratics which are the results of squaring binomials. (Remember that “trinomial” means “three-term polynomial”.) For instance:

(x + 3)2

= (x + 3)(x + 3)

x2 + 6x + 9

…so x2 + 6x + 9 is a perfect square trinomial.

Recognizing the pattern to perfect squares isn’t a make-or-break issue — these are quadratics that you can factor in the usual way — but noticing the pattern can be a time-saver occasionally, which can be helpful on timed tests.

Perfect Square Trinomial Formula

The trick to seeing this pattern is really quite simple: If the first and third terms are squares, figure out what they’re squares of. Multiply those things, multiply that product by 2, and then compare your result with the original quadratic’s middle term. If you’ve got a match (ignoring the sign), then you’ve got a perfect-square trinomial. And the original binomial that they’d squared was the sum (or difference) of the square roots of the first and third terms, together with the sign that was on the middle term of the 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.

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.

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:
Squaring a binomial creates a perfect square trinomial:
(a + b)2
(a – b)2

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

What we need to do now, is to “remember” these patterns
so that we can be on the look-out for them when factoring.

Notice the Pattern of the middle term:
The middle term is twice the product of the binomial’s first and last terms.
(a + b)² middle term +2ab
(a – b)² middle term -2ab
In (a – b), the last term is –b.

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 ax2 + bx + c is said to be perfect square, if only it satisfies the condition b2 = 4ac.

The Perfect Square Trinomial Formula is given as,


Question: Is the x– 6x + 9 a perfect square?


x2 – 6x + 9
= x2 – 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

Perfect Square Trinomial Formula

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 and you should be able to take 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) = x2 + 4x + 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 a2, b2, and 2ab

(a + b)2 = a2 + 2ab + b2

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

a = 2x
b = 3y

Step One: Square the a
a2 = 4x2

Step Two: Square the b
b2 = 9y2

Step Three: Multiply 2 by a by ‘b
2(2x)(-3y) = -12xy

Step Four: Add a2, b2, and 2ab
4x2 – 12xy + 9y2

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 algrebraic expressions containing only two terms. Example: x + 3
  • Trinomials are algebraic expressions that contain three terms. Example: 3x2 + 5x – 6

Perfect square trinomials are algebraic expressions with three terms that are created by multiplying a binomial to itself. Example: (3x + 2y)2 = 9x2 + 12xy + 4y2

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.