# (a + b - c)^2 Formula

The **(a + b - c) ^{2} formula** is used to calculate the squares of three numbers with different operations. a plus b minus c Whole Square Formula is one of the major algebraic identities and can be applied in factorization. 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 plus B minus C Whole Square 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 }= a^{2 }+ ab - ac + ab + b^{2 }- bc - ca - bc + c^{2}

(a + b - c)^{2 }= a^{2} + b^{2} + c^{2} + 2ab - 2bc - 2ca

(a + b - c)^{2 }= a^{2} + b^{2} + c^{2} + 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 plus B minus C Whole Square 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 }= a^{2} + b^{2} + c^{2} + 2ab - 2bc - 2ca

(a + b - c)^{2 } = 2^{2} + 4^{2} + 3^{2} + 2(2)(4) - 2(4)(3) - 2(3)(2)

(a + b - c)^{2 }= 4 + 16 + 9 + 16 - 24 - 12

**Answer:** (a + b - c)^{2} = 9.

**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 }= a^{2} + b^{2} + c^{2} + 2ab - 2bc - 2ca

(a + b - c)^{2 } = 12^{2} + 4^{2} + 5^{2} + 2(12)(4) -2(4)(5) - 2(5)(12)

(a + b - c)^{2 }= 144 + 16 + 25 + 96 - 40 - 120 = 121

**Answer:** (a + b - c)^{2} = 121.

**Example 3: **Find the value of a^{2} + b^{2} + c^{2} if (ab - bc - ca) = 10 and (a + b - c) = 20 using (a + b - c)^{2 }formula.

**Solution:**

To find: a^{2} + b^{2} + c^{2}

Given that:

(ab-bc-ca) = 10 and (a + b - c) = 20

Using the (a + b - c)^{2} formula,

(a + b - c)^{2 }= a^{2} + b^{2} + c^{2} + 2(ab - bc - ca)

(20)^{2 } = a^{2} + b^{2} + c^{2} + 2(10)

400 = a^{2} + b^{2} + c^{2} + 20

a^{2} + b^{2} + c^{2 }= 400 - 20 = 380

**Answer:** a^{2} + b^{2} + c^{2} = 380.

## FAQs on (a + b - c)^{2} Formulas

### What Is the Expansion of (a + b - c)^{2} Formula?

(a + b - c)^{2} formula is read as a plus b minus c whole square. Its expansion is expressed as (a + b - c)^{2 }= a^{2} + b^{2} + c^{2} + 2(ab - bc - ca).

### What Is the (a + b - c)^{2} Formula in Algebra?

The (a + b - c)^{2} formula is one of the important algebraic identities. It is read as a plus b minus c whole square. The (a + b - c)^{2} formula is expressed as (a + b - c)^{2 }= a^{2} + b^{2} + c^{2} + 2(ab - bc - ca).

### How To Simplify Numbers Using the (a + b - c)^{2} Formula?

Let us understand the use of the (a + b - c)^{2} formula with the help of the following example.

**Example:** Find the value of (2 + 5 - 3)^{2} using the (a + b - c)^{2} formula.

To find: (2 + 5 - 3)^{2}

Let us assume that a = 2 and b = 5 and c = 3.

We will substitute these in the formula of (a + b - c)^{2}.

(a + b - c)^{2 }= a^{2} + b^{2} + c^{2} + 2(ab - bc - ca)

= 2^{2} + 5^{2} + 3^{2} + 2[(2*5) - (5*3) - (3*2)]

= 4 + 25 + 9 + 2[(10) - (15) - (6)]

= 4 + 25 + 9 + 2[-11]

**Answer:** (2 + 5 - 3)^{2} = 16

### How To Use the A plus B minus C Whole Square Formula Give Steps?

The following steps are followed while using (a + b - c)^{2} formula.

- Firstly observe the pattern of the numbers whether the three numbers have ^2 as whole power or not, such as (a + b - c)
^{2}. - Write down the formula of (a + b - c)
^{2}. - (a + b - c)
^{2 }= a^{2}+ b^{2}+ c^{2}+ 2(ab - bc - ca) - Substitute the values of a, b, and c in the (a + b - c)
^{2}formula and simplify.

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