The Expansion of (a ± b ± c)^2 can be read as the whole squares of a plus or minus b plus or minus c. It is the formula that is used to find the square of sum or difference among three constant terms such as a, b, c. To derive the Expansion of Powers of Binomials and Trinomials we need to use some set of formulas. Let us learn the derivations of (a + b + c)² and (a – b – c)² with some suitable examples from this page.

Also, Read: Expansion of (a ± b)^2

## Expansion of (a ± b ± c)^2 | Expansion of Trinomial with Power 2

Expansion with three constant terms like a, b, c is known as the trinomial expressions. Check the below section to know how to derive (a + b + c)² and (a – b – c)² and where it is used.

**Derivations of (a + b + c)²: **

The derivation of (a + b + c)² is explained in detail here.

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

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

(a + b + c)² = a² + ab + ac + ab + b² + bc + ac + bc + c²

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

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

Thus the formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

**Derivation of (a – b – c)²:**

The derivation of the trinomial expression (a – b – c)² is explained in detail here.

(a – b – c)² = (a – b – c) (a – b – c)

(a – b – c)² = a (a – b – c) – b (a – b – c) -c (a – b – c)

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

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

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

Thus the formula of (a – b – c)² = a² + b² + c² – 2(ab – bc + ca)

### Expansion of (a ± b ± c)^2 Examples

Let us see some suitable examples on the expansion of (a ± b ± c)² to know how and where to use the formulas of (a + b + c)² and (a – b – c)² in algebra.

**Example 1.**

Find the value of (a + b + c)² if the values of a, b, c are 3, 4, 2.

**Solution:**

Given the values of a, b, c are 3, 4, 2.

We have to substitute the values of a, b, c in the formula (a + b + c)²

We know that

The formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(3 + 4 + 2)² = 3² + 4² + 2² + 2(3(4) + (4)2 + 2(3))

= 9 + 16 + 4 + 2(12 + 8 + 6)

= 29 + 2(26)

= 29 + 52

= 81

**Example 2.**

Expand (2a + 4b + 3c)² by using the (a + b + c)² formula.

**Solution:**

Given the trinomial expression (2a + 4b + 3c)²

We know that

The formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

We have to substitute the values of a, b, c in the formula (a + b + c)²

(2a + 4b + 3c)² = (2a)² + (4b)² + (3c)² + 2((2a)(4b) + (4b)3c + 3c(2a))

(2a + 4b + 3c)² = 4a² + 16b² + 9c² + 2(8ab + 12bc + 6ac)

(2a + 4b + 3c)² = 4a² + 16b² + 9c² + 16ab + 24bc + 12ac

Thus the expansion of (2a + 4b + 3c)² is 4a² + 16b² + 9c² + 16ab + 24bc + 12ac

**Example 3.**

Find the value of (a – b – c)² if the values of a, b, c are 5, 3, 2.

**Solution:**

Given the values of a, b, c are 5, 3, 2.

We know that

The formula of (a – b – c)² = a² + b² + c² – 2(ab – bc + ca)

We have to substitute the values of a, b, c in the formula (a – b – c)²

(5 – 3 – 2)² = 5² + 3² + 2² – 2((5)(3) – (3) (2) + (2)(5))

(5 – 3 – 2)² = 25 + 9 + 4 – 2(15 – 6 + 10)

(5 – 3 – 2)² = 38 – 2(19)

(5 – 3 – 2)² = 38 – 38 = 0

Thus the value of (a – b – c)² if the values of a, b, c are 5, 3, 2 is 0.

**Example 4.**

Expand (3x + 2y + 5z)² by using the (a + b + c)² formula.

**Solution:**

Given the trinomial expression (3x + 2y + 5z)²

We know that

The formula of (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

We have to substitute the values of a, b, c in the formula (a + b + c)²

(3x + 2y + 5z)² = (3x)² + (2y)² + (5z)² + 2((3x)(2y) + (2y)(5z) + (5z)(3x))

(3x + 2y + 5z)² = 9x² + 4y² + 25z² + 2(6xy + 10yz + 15zx)

(3x + 2y + 5z)² = 9x² + 4y² + 25z² + 12xy + 20yz + 30zx

Thus the expansion of (3x + 2y + 5z)² is 9x² + 4y² + 25z² + 12xy + 20yz + 30zx

**Example 5.**

Expand (8x – 4y – 6z)² by using the (a – b – c)² formula.

**Solution:**

Given the trinomial expression (8x – 4y – 6z)²

We know that

The formula of (a – b – c)² = a² + b² + c² – 2(ab – bc + ca)

We have to substitute the values of a, b, c in the formula (a – b – c)²

(8x – 4y – 6z)² = (8x)² + (4y)² + (6z)² – 2((8x)(4y) – (4y)(6z) + (6z)(8x))

(8x – 4y – 6z)² = 64x² + 16y² + 36z² – 2(32xy) – 24yz + 48zx)

(8x – 4y – 6z)² = 64x² + 16y² + 36z² – 64xy – 48yz + 96zx

Thus the expansion of (8x – 4y – 6z)² is 64x² + 16y² + 36z² – 64xy – 48yz + 96zx

### FAQs on Expansion of (a ± b ± c)²

**1. What is the Expansion of the (a – b – c)² Formula?**

The Expansion of (a – b – c)² formula is a² + b² + c² – 2(ab – bc + ca) or a² + b² + c² – 2ab + 2bc – 2ac

**2. What is the Expansion of (a + b + c)² Formula?**

The Expansion of (a + b + c)² formula is a² + b² + c² + 2(ab + bc + ca) or a² + b² + c² + 2ab + 2bc + 2ac

**3. What type of expression is Expansion of (a ± b ± c)^2?**

The Expansion of (a ± b ± c)^2 is a trinomial expression because it consists of three constant terms.