a plus b minus c Whole Square

a plus b minus c whole square, i.e., (a + b – c)² is equal to [a² + b² + c² + 2(ab – bc – ca)]. It is represented as (a + b – c)². The Square of a number is defined as the product of the number itself. Hence, the value of (a + b – c) is (a + b – c) x (a + b – c).

In this article, we will learn what the value of a plus b minus c whole square or (a + b – c)², how to find its value how to apply it in problems, and its applications. Let’s start our learning on the topic (a + b – c)².

What is (a + b – c)² Expression

a plus b minus c whole square or (a + b – c)² is an algebraic expression where a, b, c are the algebraic terms that represent numbers or values. Here we find the sum of a, b, then the difference of the sum found and c, the square of the result obtained is the final result. More simply (a + b – c)² is obtained by the sum of squares of all the three individual terms and the twice of all the products of two terms i.e., ab, -bc and -ca.

(a + b – c)² Formula

Below is the formula for (a + b – c)²

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

Expanding Expression

Expression (a + b – c)² can be expanded easily by breaking it as a product of (a + b – c) with itself or by using algebraic identity (x+y)² and (x-y)². The expanded form of expression (a + b – c)² is shown in the image added below:

Expression (a + b – c)²

Let’s discuss these methods in detail as follows:

Product of (a + b – c) with itself

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

= a.(a+b-c) + b.(a+b-c) – c.(a+b-c)

= a²+ab-ac+ab+b²-bc-ac-bc+c²

= a² + b² + c² + 2( ab – bc – ca ) {Collecting like terms}

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

Using Algebraic Identities

• (x-y)² = x²+y²-2xy…(i)

• (x+y)² = x²+y²+2xy…(ii)

Let, x = a+b and y = c

(a + b – c)² = (a+b)² + c²– 2(a+b)(c) {By (i)}

= (a²+b²+2ab) +c² -2(ac+bc) {By (ii)}

= a² +b² +2ab +c² -2ac -2bc

= a² + b² + c² + 2( ab – bc – ca) {Collecting like terms}

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

Applications of (a + b – c)²

Square of the quantity (a + b – c) is expressed as (a + b – c)². Numerous applications of this statement can be found in science, engineering, and mathematics. Here are a few instances:

• Algebra: This expression can be used to expand and simplify variable-based statements in algebra. One way to get a² + b² + c² + 2ab – 2ac – 2bc is to extend (a + b – c)².

• Geometry: The area of several geometric shapes can be found using the formula (a + b – c)². For example, it can be useful to determine the area of a square or rectangle whose sides are given by the terms a, b, and c.

• Statistics: In statistics, this expression could be used in the context of calculating variances or deviations from a mean value, particularly when dealing with multiple variables.

• Physics: In energy-related equations, such as kinetic energy or potential energy formulations, where the terms indicate various contributing components, (a + b – c)² can appear.

• Computer Science: Expressions such as (a + b – c)² may be found in algorithms in computer science, especially in numerical approaches or optimizations.

Conclusion

(a + b – c)² is a simple algebraic expression that is equal to a² + b² + c² + 2( ab – bc – ca ) and is very useful in simplifying expressions in mathematical aspects. It finds its application in Algebra, Geometry, Statistics ,Physics, Computer Science , Engineering and Finance.

Examples on (a + b – c)²

Example 1: Evaluate: (A + 5B – 2C)².

Solution:

For (A + 5B – 2C)²

a = A, b = 5B and c = 2C,

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

(A + 5B – 2C)² = (A)² + (5B)² + (2C)² + 2(A)(5B) + 2(5B)(-2C) + 2(A)(-2C)

⇒ (A + 5B + 2C)² = A² + 25B² + 4C² + 10AB – 20BC – 4AC

Example 2: Find the value of (a + b – c)² if a = 2, b = 2, and c = 4.

Solution:

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

= 2²+2²+4² + 2(2)(2)-2(2)(4)-2(4)(2)

= 4 + 4 + 16 + 8 – 16 – 16

= 16 + 16 – 16 – 16 = 0

Example 3: Find the value of (a + b – c)² if a = 2, b = 3, and c = 5.

Solution:

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

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

= 4+9+25+12-30-20

= 50 – 50 = 0

Practice Questions

Q1: Evaluate: (5x + 3y – 4z)².

Q2: Find the area of a square whose sides are expressed as (a+b-c) and a=2,b=4,c=10.

Q3: Find the value of (a + b – c) if a² + b² + c² = 5 and (ab – bc – ac) = 11.

Q4: Find the value of (a² + b² + c²) if (a + b – c) = 10 and (ab – bc – ac) = 20.

Q5: Find the square of the expression (a+b-c) where a = b = c = 5.

Frequently Asked Questions

What is formula for (a + b – c) whole square?

General formula for (a + b – c) whole square is given by: (a + b – c)² = a² + b² + c² + 2(ab – bc – ac)

How is (a + b – c)² different from (a – c + b)²?

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

It is just a rearranged formula but the expanded formula gives the same result. Hence there is no difference between (a + b – c)² and (a – b + c)².

What are some practical examples of (a + b – c)² in daily life?

To find the area of a square with sides as (a+b-c) where a,b,c are some values ,we can use the (a+b-c)² formula.

Can (a + b – c)² be applied in computer science or programming? 

Yes, in computer science or programming (a + b – c)² may be applied in algorithms , especially in numerical approaches or optimizations.