Square of a Trinomial

Square of a Trinomial Expansion | Perfect Square Trinomial Definition, Formula & Solved Examples

Do you want to expand trinomials easily without any confusion and hassle? You should refer to this page. Here, we have explained how to expand the Square of a Trinomial and perfect square trinomial definition and formulas. Students who need more subject knowledge about square trinomials and solve any kind of trinomial expansions must go with this article completely. In this article, you will also get some worked-out examples on Square of a Trinomial and Perfect square trinomial. So, let’s continue your read and learn the concept of square trinomial.

Perfect Square Trinomial Definition & Formula

An expression obtained from the square of the binomial equation is a perfect square trinomial. When the trinomial is in the form ax² + bx + c then it is said to be a perfect square, if and only if it meets the condition b² = 4ac.

The Perfect Square Trinomial Formula is as follows,

(ax)²+2abx+b² = (ax+b)²
(ax)²−2abx+b² = (ax−b)²

How to Expand the Square of a Trinomial?

Here, we are discussing the expansion of the square of a trinomial (a + b + c).

Let (b + c) = x

(i) Then (a + b + c)2 = (a + x)2 = a2 + 2ax + x2
= a2 + 2a (b + c) + (b + c)2
= a2 + 2ab + 2ac + (b2 + c2 + 2bc)
= a2 + b2 + c2 + 2ab + 2bc + 2ca

Therefore, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.

(ii) (a – b – c)2 = [a + (-b) + (-c)]2
= a2 + (-b2) + (-c2) + 2 (a) (-b) + 2 (-b) (-c) + 2 (-c) (a)
= a2 + b2 + c2 – 2ab + 2bc – 2ca

Therefore, (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca.

(iii) (a + b – c)2 = [a + b + (-c)]2
= a2 + b2 + (-c)2 + 2ab + 2 (b) (-c) + 2 (-c) (a)
= a2 + b2 + c2 + 2ab – 2bc – 2ca

Therefore, (a + b – c)2 = a2 + b2 + c2 + 2ab – 2bc – 2ca.

(iv) (a – b + c)2 = [a + (- b) + c]2

= a2 + (-b2) + c2 + 2 (a) (-b) + 2 (-b) (-c) + 2 (c) (a)
= a2 + b2 + c2 – 2ab – 2bc + 2ca

Therefore, (a – b + c)2 = a2 + b2 + c2 – 2ab – 2bc + 2ca.

Solved Examples on Square Trinomial

1. Expand (x+4y+6z)2

Solution:

Given trinomial expression is (x+4y+6z)2

We know that (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.

Here a=1x, b=4y, c=6z

Now, substitute the values in the expression of (a + b + c)2

Then (x+4y+6z)2 = (1x)2 + (4y)2 + (6z)2 + 2(1x)(4y) + 2(4y)(6z) + 2(6z)(1x)

= x2 + 16y2 + 36z2 + 8xy + 48yz + 12zx

Hence, (x+4y+6z)2 = x2 + 16y2 + 36z2 + 8xy + 48yz + 12zx. 

2. Is the trinomial x– 6x + 9 a perfect square?

Solution:

Given trinomial is x2 – 6x + 9, now calculate the expression and find it is a perfect square or not.

x2 – 6x + 9 = x2 – 3x – 3x + 9
= x(x – 3) – 3(x – 3)
= (x – 3)(x – 3)

Otherwise,

x2 – 6x + 9 = x2 – 2(3)(x) + 32 = (x – 3)2

The factors of the given equation are a perfect square.

Therefore, the given trinomial is a perfect square.

3. Simplify a + b + c = 16 and ab + bc + ca = 40. Find the value of a2 + b2 + c2.

Solution:

As per the given question, a + b + c = 16

Now, by squaring both sides, we get

(a+ b + c)2 = (16)2

a2 + b2 + c2 + 2ab + 2bc + 2ca = 256

a2 + b2 + c2 + 2(ab + bc + ca) = 256

a2 + b2 + c2 + 2 × 40 = 256 [Given, ab + bc + ca = 40]

a2 + b2 + c2 + 80 = 256

At this step, we have to subtract 80 from both sides

a2 + b2 + c2 + 80 – 80 = 256 – 80

a2 + b2 + c2 = 176

Hence, the square of a trinomial formula will help us to expand and get the result for a2 + b2 + c2 is 176. 

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