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Perfect Square Trinomial

Perfect Square Trinomial is the product of two binomials. But, both the binomials are same. When factoring some quadratics which gives identical factors, that quadratics are Perfect Square Trinomial.

The general forms of perfect square trinomial are
(ax - b) 2 = (ax)2 - 2axb + b2 and (ax + b) 2= (ax)2 + 2axb + b2

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Perfect Square Trinomial Definition

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A perfect square trinomial is one that factors into two identical factors. A trinomial of the form a2 + 2ab + b2 or a2 - 2ab + b2 is a perfect square trinomial with factors (a + b)2 or (a - b)2 respectively.


The trinomial in the form ax2 + bx + c is perfect squre trinomial, if b2 = 4ac.

Perfect Square Trinomial Formula

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Below you can see the formulas for perfect square trinomials:
a2 + 2ab + b2 = (a + b)2

a2 - 2ab + b2 = (a - b)2

Solved Example

Question: Factor using the perfect square trinomial formula x2 + 14x + 49.
Solution:
Given x2 + 14x + 49

The first and last terms are perfect squares.

=> x2 + 14x + 49 = x2 + 14x + 72

The middle term is twice the product of the value x and the value 7.

=> 14x = 2 * 7x

Therefore, we can use the formula, a2 + 2ab + b2 = (a + b)2

Thus, x2 + 14x + 49 = (x + 7)(x + 7) = (x + 7)2


Factoring Perfect Square Trinomials

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Perfect Square Trinomial is the product of two same binomials. In this, the first term and last term of the perfect square are perfect squares and the middle term is 2 times the square root of first terms times and square root of last terms.

Solved Examples

Question 1: Factor the trinomial x2 + 4x + 4
Solution:

Given x2 + 4x + 4

The above trinomial can be written as x2 + 2x + 2x + 4

Take x as common from first two terms,

x(x + 2) + 2x + 4

Take 2 as common from last two terms.

x(x + 2) + 2(x + 2)

(x + 2)(x + 2)

In this it is the product of two identical binomials

(x + 2)2 , which is a perfect square.



Question 2: Factor the trinomial x2 + 8x + 16
Solution:

Given x2 + 8x + 16

The above trinomial can be written as x2 + 4x + 4x + 16

Take x as common from first two terms,

x(x + 4) + 4x + 16

Take 4 as common from last two terms.

x(x + 4) + 4(x + 4)

(x + 4)(x + 4)

In this, it is the product of two identical binomials

(x + 4)2, which is a perfect square.



Perfect Square Trinomial Examples

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Given below are some of the examples on perfect square trinomials.

Solved Examples

Question 1: Factor the trinomial x2 + 4x + 4
Solution:
Given trinomial x2 + 4x + 4

In this, the first term is x2, the second term is 2 times the square root of first term ($\sqrt{x^2}$ = x) and square root of second term ($\sqrt{4}$ = 2) and the third term 4 can be written as 22 which is a perfect square.

x2 + 4x + 4 = x2 + 2 * 2 * x + 4 = (x + 2)(x + 2) = (x + 2)2.

This is the perfect square trinomial.



Question 2: Factor the trinomial x2 + 8x + 16
Solution:
Given x2 + 8x + 16

In this, the first term is x2, the second term is 2 times the square root of first term ($\sqrt{x^2}$ = x) and square root of second term ($\sqrt{16}$ = 4) and the third term 16 can be written as 42 which is a perfect square.

x2 + 8x + 16 = x2 + 2 * 4 * x + 42 = (x + 4)(x + 4) = (x + 4)2.

This is the perfect square trinomial.



Question 3: Factor the trinomial x2 + 10x + 25
Solution:
Given x2 - 10x + 25
In this, the first term is x2, the second term is -2 times the square root of first term ($\sqrt{x^2}$ = x) and square root of second term ($\sqrt{25}$ = 5) and the third term 25 can be written as 52 which is a perfect square.

x2 - 10x + 25 = x2 - 2 * 5 * x + 52 = (x - 5)(x - 5) = (x - 5)2.

This is the perfect square trinomial.



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