The perfect square is a number that is obtained by multiplying the number by itself. Similarly the perfect square trinomial is an algebraic expression that is obtained by multiplying the two same binomials. A trinomial is an expression that consists of three terms whereas binomial consists of two terms i.e. a variable and constant which are separated either by the “+” or “-” sign. Even in trinomial too the terms are separated either by the “+” or “-” sign. Generally perfect square trinomial exists in two forms. i.e. a² + 2ab + b² or a² – 2ab + b². Now let’s look into the steps to find the perfect square trinomial from the given binomial and vice versa.

Perfect Square Trinomial Formula

We can determine whether the given trinomial is a perfect square trinomial or not by a simple formula. Let’s consider a trinomial ax² + bx + c where x is a variable and a, b, c are constants then the given trinomial is a perfect square trinomial if and only if it satisfies the below condition

b² – 4ac = 0

a, b, c are constants.

Steps to find perfect square trinomial from binomial

So, the first term in the trinomial is the square of first term in the binomial. The second term is twice the product of two terms in the binomial. The third term in the trinomial is the square of second term in the binomial. If the binomial has a positive sign then all the terms in the trinomial formed by squaring the binomial are positive. If the binomial has a negative sign then the second term in the perfect squared trinomial will have negative sign.

(a + b)² = a² + 2ab + b²

(a – b)² = a² – 2ab + b²

Here a, b are first and second terms in a binomial.

Below are the basic steps that are needed to be followed to find the perfect square trinomial from binomial,

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

Steps to factorize the Perfect Square Trinomial

1. Write the perfect square trinomial in the form a² + 2ab + b² or a² – 2ab + b² where the first & third terms are perfect squares, one being a variable and the other being constant.
2. Also, the middle term is twice the product of the first and third terms.
3. If the middle term has a positive sign then factors of a trinomial are (a+b)(a+b). Else (a-b)(a-b).

a² + 2ab + b² = (a + b)²

a² – 2ab + b² = (a – b)²

Sample Questions

Question 1: Check the given trinomial x² + 4x + 4 is perfect square trinomial or not.

Solution:

Given trinomial, x² + 4x + 4

On comparing given trinomial with ax² + bx + c

a = 1, b = 4, c = 4

b² – 4ac = 4² – 4 × 1 × 4

= 16 – 16

= 0

b² – 4ac = 0, So given trinomial is perfect square trinomial.

Question 2: Check the given trinomial x² + 3x – 2 is perfect square trinomial or not.

Solution:

Given trinomial, x² + 3x – 2

On comparing given trinomial with ax² + bx + c

a = 1, b = 3, c = -2

b² – 4ac = 3² – 4 × 1 × 2

= 9 – 8

= 1

b² – 4ac ≠ 0, So given trinomial is not a perfect square trinomial.

Question 3: Find the factors of the given perfect square trinomial x² – 6x + 9.

Solution:

Given trinomial, x² – 6x + 9

It can be rewritten as x² – 2(3)x + 3² which is of form a² – 2ab + b² so factors are (a – b)(a – b)

Where a = x, b = 3

So, factors of given perfect square trinomial are (x – 3)(x – 3)

Question 4: Find the factors of the given perfect square trinomial x² + 8x + 16.

Solution:

Given trinomial, x² + 8x + 16

It can be rewritten as x² + 2(4)x + 4² which is of form a² + 2ab + b² so factors are (a + b)(a + b)

Where a = x, b = 4

So, factors of given perfect square trinomial are (x + 4)(x + 4)

Question 5: Find the perfect square trinomial for the binomial (x + 5).

Solution:

Given binomial, x + 5

The perfect square trinomial for a binomial of form a + b is a² + 2ab + b²

From the given polynomial a = x, b = 5

a² + 2ab + b² = x² + 2(5)x + 5²

= x² + 10x + 25

So, x² + 10x + 25  is the perfect square trinomial for the given binomial.

Question 6: Find the perfect square trinomial for the binomial (2x – 1).

Solution:

Given binomial, 2x – 1

The perfect square trinomial for a binomial of form a – b is a² – 2ab + b²

From the given polynomial a = 2x, b = 1

a² – 2ab + b² = (2x)² – 2(1)(2x) + 1²

= 4x² – 4x + 1

So, 4x² – 4x + 1 is the perfect square trinomial for the given binomial.