# Why are Algebraic Expressions Useful?

Algebraic expression started in the 9th century. In the beginning, it was more in statement form and not mathematical at all. For instance, algebraic equations used to be written as “5 times the thing added with 3 gives 18” which is basically 5x + 3 = 18. This type of equation which was not mathematical was Babylonian algebra. Algebra evolved with time and with the different forms provided. It started with Egyptian algebra, then came Babylonian algebra, then came greek geometrical algebra, moved to diophantine algebra, followed by Hindu algebra, then came Arabic algebra, and followed by abstract algebra. Today, the easiest and most convenient form of algebra is taught in classes for better understanding.

### Algebraic Expressions

Algebraic expressions are the expressions obtained from the combination of variables, constants, and mathematical operations like addition, subtraction, multiplication, division, and so on. An algebraic expression is made up of terms, there can be one or more than one term present in the equation. Let’s learn about the basic terms used in algebraic expressions,

### Constants, Variables, Coefficients, and Terms

In the algebraic expression, fixed numerical are called constants, constants do not have any variables attached to them. For example, 3x – 1 has a constant -1 to it. Variables are the unknown values that are present in the algebraic expression, for instance, 4y + 5z has y and z as variables. Coefficients are the fixed values (real numbers) attached to the variables, they are multiplied with the variables. For example, 5x^{2} + 3 has 5 as the coefficient of x^{2}. A term can be a constant, a variable, or a combination of both, basically, each term is separated by either addition or subtraction. For example, 3x + 5, 3x and 5 are the terms.

### Algebraic Expressions Formulas

There are basic algebraic formulae that are used in mathematics. Then there are four algebraic identities that are used, identities are those fixed equations that are true under all conditions. Let’s take a look at the fixed identities,

- (a + b)
^{2}= a^{2}+ b^{2}+ 2ab - (a – b)
^{2}= a^{2 }+ b^{2}– 2ab - (a + b)(a – b) = a
^{2}– b^{2} - (x + a)(x + b) = x
^{2}+ 2(a + b) + ab

Now, let’s look at the algebraic expression formulae and see some examples based on those formulae, these formulae contain three variables and the exponents go up to 3.

- (a + b)
^{2}= a^{2}+ b^{2}+ 2ab - (a – b)
^{2}= a^{2 }+ b^{2}– 2ab - (a + b)(a – b) = a
^{2}– b^{2} - (a + b)
^{3}= a^{3 }+ 3ab(a + b) + b^{3} - (a – b)
^{3}= a^{3}– 3ab(a – b) – b^{3} - a
^{3}+ b^{3}= (a + b)(a^{2}+ ab + b^{2}) - a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 2ab + 2bc + 2ac

### Why are Algebraic Expressions Useful?

Algebraic expressions not only are valuable in mathematics but also extremely important in real life. People do not often realize when and where they are using algebraic expressions but they are in some or the other way are a part of this creation in mathematics. One need not be a mathematician to understand, acknowledge and use algebra in real life. For instance, Once a shopkeeper said to the customer that he will be providing the customer with 5 fewer bananas than he has in his shop. The shop has 12 bananas. Now, the customer will use basic maths and identify that the number of bananas gained are 12 – 5 = 7. However, it is interesting to see that algebra is used here. The number of bananas obtained by the customer was unknown, hence name it “x”, the number of bananas shopkeeper had was 12 and 5 less than what shop has was to be given to the customer. Therefore, a simple algebraic equation with variable and constant is formed, x + 5 = 12, x = 7. Let’s take a look at some points pointing out the importance of algebraic expressions and their usefulness,

- Algebraic expressions are used in solving different and complex equations in mathematics.
- Algebraic expressions can be seen in computer programming, for example, they are used for tasks of inference.
- Algebraic expressions are used in economics to find out the revenue, cost, etc.
- In different fields of mathematics, for instance, trigonometry, geometry, etc. algebraic expressions are required to solve the unknown angles and values.
- Algebra is useful in boosting one’s logical reasoning ability and aptitude.
- Making important and significant decisions in maths and in real life becomes easy if one has good knowledge of algebra.

### Sample Problems

**Question 1: Find out the value of the term, (2 + 3) ^{2} using algebraic formulae.**

**Solution:**

Using the algebraic formula,

(a + b)

^{2}= a^{2}+ b^{2}+ 2ab(2 + 3)

^{2}= 2^{2}+ 3^{2}+ 2 × 2 × 3(2 + 3)

^{2}= 4 + 9 + 12(2 + 3)

^{2}= 25

**Question 2: Find out the value of the term, (5 – 3) ^{2} using algebraic formulae.**

**Solution:**

Using the algebraic formula,

(a – b)

^{2}= a^{2}+ b^{2}– 2ab(5 – 3)

^{2}= 5^{2}+ 3^{2}– 2 × 5 × 3(5 – 3)

^{2}= 25 + 9 – 30(5 – 3)

^{2}= 4

**Question 3: A shopkeeper has 45 books, two customers walked in to get books. Customer one wanted double of what customer second wanted. Customer two wanted 30 less than what the shop has. How many books each does the customer want?**

**Solution:**

Suppose customer one wants “x” number of books and customer two wants “y” number of books,

Given:

x = 2y

y = 45 – 30

y = 15

Customer second wants 15 books.

x = 2 × 15

x = 30

Customer one wants 30 books.

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