(a - b - c)^2 Formula

The (a - b - c)2 formula is used to find the square of the difference between the three numbers without actually calculating the whole square and in factorization. (a - b - c)2 formula is one of the major algebraic identities. To derive the expansion of (a - b - c)2 formula we just multiply (a - b - c) by itself to get (a - b - c)2. Let us learn more about the (a - b - c)2 formula along with solved examples.

What Is (a - b - c)2 Formula?

We just read that by multiplying (a - b - c) by itself we can easily derive the (a - b - c)2 formula. Let us see the expansion of (a - b - c)2 formula. (a - b - c)2 = (a - b - c)(a - b - c) (a - b - c)2 = a2 - ab - ac - ab + b2 + bc - ca + bc + c2 (a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ca (a - b - c)2 = a2 + b2 + c2 - 2(ab - bc + ca)

Let us see how to use the (a - b - c)2 formula in the following section.

Examples on (a - b - c)2 Formula

Let us take a look at a few examples to better understand the formula of (a - b - c)2.

Example 1: Find the value of (a - b - c)2 if a = 2, b = 4, and c = 3 using (a - b - c)2 formula.

Solution:

To find: (a - b - c)2 Given that: a = 2, b = 4, c = 3 Using the (a - b - c)2 formula, (a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ca (a - b - c)2 = 22 + 42 + 32 - 2(2)(4) +2(4)(3) - 2(3)(2) (a - b - c)2 = 4 + 16 + 9 - 16 + 24 - 12

Answer: (a - b - c)2 = 25.

Example 2: Find the value of (a - b - c)2 if a = 12, b = 4, and c = 5 using (a - b - c)2 formula.

Solution:

To find: (a - b - c)2 Given that: a = 12, b = 4, c = 5 Using the (a - b - c)2 formula, (a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ca (a - b - c)2 = 122 + 42 + 52 - 2(12)(4) +2(4)(5) - 2(5)(12) (a - b - c)2 = 144 + 16 + 25 - 96 + 40 - 120 = 9

Answer: (a - b - c)2 = 9.

Example 3: Find the value of a2 + b2 + c2 if (ab - bc + ca) = 10 and (a - b - c) = 20 using (a - b - c)2 formula.

Solution:

To find: a2 + b2 + c2 Given that: (ab-bc+ca) = 10 and (a - b - c) = 20 Using the (a - b - c)2 formula, (a - b - c)2 = a2 + b2 + c2 - 2(ab - bc + ca) (20)2 = a2 + b2 + c2 - 2(10) 400 = a2 + b2 + c2 - 20 a2 + b2 + c2 = 400 + 20 = 420

Answer: (a - b - c)2 = 420.