Perfect Square Trinomial Formula

Perfect Square Trinomial is when a binomial is multiplied with itself it gives an expression which consists of three terms and this expression. Perfect square trinomials consist of variable as well as constant terms in their algebraic expression. It can be represented in the form of ax² + bx +c where x is a variable and a, b, c are real numbers.

Example: (x + 2) × (x + 2) = x² + 4x + 4 is a perfect square trinomial where (y + 2) is binomial expression.

Now let’s understand the binomial, so binomials are arithmetic expressions that are formed by combining two terms by a positive sign or by a negative sign.

Example: (x + 5), (x – 5) are binomial expressions.

Properties of Perfect Square Trinomial

• It is formed by the multiplication of binomials with themselves.
• It can be represented by the formula ax² + bx+c where a, b, c are real numbers provided a should not be equal to 0.
• It can be solved using two identities (a + b)² = (a² + 2×a×b + b²) and (a – b)² = (a² -2×a×b +b²)
• It can be solved to get two equal binomials.

Perfect Square Trinomial Formula

The formula for Perfect Square Trinomial is given by two expressions

(ax + b)² = (ax)² + 2×(ax)×(b) + b²

Example:

Lets take expression x² + 10x + 25

According to above formula  (ax)² = x² so a = 1

b² = 25 so b = 5

2×(ax)×(b) = 10×x which is true

Hence the given expression is a Perfect Square Trinomial and can be decomposed to binomial expression by using the above formula.

So (x + 5)² = x² + 10x + 25

(ax – b)² = (ax)² – 2×(ax)×(b) + b²

Example:

Lets take expression x² – 10x + 25

According to above formula  (ax)² = x² so a = 1

b² = 25 so b = 5

2×(ax)×(b) = 10×x which is true

Hence the given expression is a Perfect Square Trinomial and can be decomposed to binomial expression by using the above formula.

So (x – 5)² = x² – 10x + 25

Sample Questions

Question 1: Find factors of the perfect square trinomial for the algebraic expression x² + 6x + 9.

Solution:

For the given algebraic expression x² + 6x + 9 

It is clear that it can be represented in the form (ax)² + 2×(ax)×b + b². 

So factors of the equation ((ax)² + 2axb + b²) are (ax+b) and (ax+b).

Here, a = 1

b² = 9 so b = 3

and 2axb = 6x.

Therefore, the factors are (x + 3) (x + 3).

Question 2: Find factors of the perfect square trinomial for the algebraic expression 9x² + 24x + 16.

Solution:

For the given algebraic expression 9x² + 24x + 16

It is clear that it can be represented in the form (ax)² + 2×(ax)×b + b².

So factors of the equation ((ax)² + 2axb + b²) are (ax+b) and (ax+b).

Here, (ax)² = 9x²

so a = 3

b² = 16 so b = 4

and 2axb = 24x.

Therefore, the factors are (3x + 4) (3x + 4).

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

Solution:

For the given algebraic expression x² – 6x + 9

It is clear that it can be represented in the form (ax)² – 2×(ax)×b + b².

So factors of the equation ((ax)² – 2axb + b²) are (ax-b) and (ax-b).

Here, a = 1

b² = 9 so b = 3

and 2axb = 6x.

Therefore, the factors are (x – 3) (x – 3).

Question 4: Find factors of the perfect square trinomial for the algebraic expression 9x² – 24x + 16.

Solution:

For the given algebraic expression 9x² – 24x + 16

It is clear that it can be represented in the form (ax)² + 2×(ax)×b + b².

So factors of the equation ((ax)² – 2axb + b2) are (ax-b) and (ax-b).

Here, (ax)² = 9x²

so a = 3

b² = 16 so b = 4

and 2axb = 24x

Therefore, the factors are (3x – 4) (3x – 4).

Question 5: Find factors of the perfect square trinomial for the algebraic expression 4x² + 12x + 9.

Solution:

For the given algebraic expression 4x² + 12x + 9

It is clear that it can be represented in the form (ax)² + 2×(ax)×b + b².

So factors of the equation ((ax)² + 2axb + b²) are (ax+b) and (ax+b).

Here, (ax)² = 4x²

so a = 2

b² = 9 so b = 3

and 2axb = 12x.

Therefore, the factors are (2x + 3) (2x + 3).