A perfect square trinomial is an expression that results when a binomial is multiplied by itself. This expression consists of three terms and can be written in the form ax2+bx+c, where x is a variable, and a, b, and c are real numbers.
Example: (x + 2) × (x + 2) = x²2 + 4x + 4 is a perfect square trinomial where (x + 2) is a binomial expression.
The formula for a Perfect Square Trinomial is given by two expressions
(ax + b)2 = (ax)2 + 2×(ax)×(b) + b2
Example 1:
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 2:
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 4x²2 + 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).
Practice Problem Based on Perfect Square Trinomial Formula
Question 1. Factor the expression 16x2−8x+1 completely, if it is a perfect square trinomial.
Question 2. Expand the following expression using the perfect square formula: (3x+7)2
Question 3. For the expression 4x2−12x+9, identify the values of a and b in the perfect square trinomial. Then, write it as the square of a binomial.
Question 4. Is the expression x2−14x+49 a perfect square trinomial? If yes, expand the binomial expression that gives this trinomial.
Answer:-
- (4x-1)2
- 9x2+42x+49
- a=2, b=3, and (2x-3)2
- (x−7)2