Basics of Algebra: Types, Formula, Questions

Basics of Algebra is an important topic to be covered in competitive exams preparation be it for IBPS Clerk, PO, RRB, SO, or SBI Clerk or PO Exam. The number of questions and the complexity of questions vary from exam to exam every year.

Basics of Algebra is complex in nature and requires an understanding of Numbers, Squares and Square roots, Cubes and Cube roots, BODMAS Rule, and Math Tables. Example questions on Basics of Algebra and practice questions on Basics of Algebra are given in this article.

Nowadays, questions in exams are mixed with multiple concepts and it requires practice and a deep understanding of the Basics of Algebra concepts along with quickly identifying numbers.

What are the Basics of Algebra?

Algebra is the most common form to study Mathematical representation and rules for their manipulation. These representations are alphabetic or symbolic and don’t have any fixed values. Well, a known example is the use of symbols such as ‘x’, ‘y’, and ‘z’.

It allows substituent values to solve the equation and through which we can find values of the equation. An algebraic expression has a variable and constant part, variable will take multiple values.

For example: 3x + 5 is an algebraic expression in which x can hold multiple values and for each value, this expression will give different results. Let’s substitute different values of x and find the final value of the expression:
For x = 1, then 3 x 1 + 5 = 08
For x = 2, then 3 x 2 + 5 = 11
For x = 3, then 3 x 3 + 5 = 14

Types of Expressions

There are the following types of Algebraic expression:

1. Monomials: An algebraic expression that has only one variable term. For Example: 5x and 5xy.
2. Binomials: An algebraic expression that has two, unlike variable terms. For Example: 5x + 5y, a + b and x² – y²
3. Trinomials: An algebraic expression that has three, unlike variable terms. For Example: 5x + 5y – 6yz, a + b – 10 and x² – y² + 6
4. Polynomials: An algebraic expression with one or more terms in it is called a polynomial. Thus above examples of monomial, binomial and trinomial are all polynomials.

Addition and Subtraction

Addition and Subtraction of two algebraic expressions are done between two equal terms. Unlike terms cannot be added or subtracted.

For Example: Adding two expression: 3x + 11 and 7x -10
As given the same terms will be added. So 3x and 7x will be added and similarly, 11 and -10 will be added. So final expression will be 10x + (11 – 10) = 10x + 1

Multiplication

Multiplication of Monomial terms will only result in Monomial expression. If we multiply Polynomial with Polynomial will result in the polynomial.

For Example: 5x multiplied by 10x. We see both expressions are Monomial. For the multiplication of this expression, the result will be monomial. 5x x 10x = 5 x 10 = 50 and x multiply by x will result in x². Hence final expression will be 50x².

Formula

(x + y)² = x² + 2 xy + y²

(x – y)² = x² – 2 xy + y²

(x + y)(x – y) = x² – y²

(x + a)(x + b) = x² + (a + b)x + y²

Finding Value of Expression

An Algebraic expression can be solved by substituting values of the placeholder or variable. Putting values in expression help in finding various result set and finding out the best suitable value for the expression.

For example: A well know formula to calculate speed is time multiplied by distance. An expression for this will be: speed = time (t) x distance (d), where t and d are placeholders or variables. Now substituting different values of t and d will result in different speeds.

Solved Examples on Basics of Algebra

Below are the few Solved Examples on Basics of Algebra:

Example 1: Find the value of x in expression 4x – 38, which will result in a positive number.

Solution 1: To find the solution to the given expression we’ll try substituting different values of x and see for which value we’re getting positive results.
Let’s try for x = 2 => 4 x 2 – 38 = -30
From the maths table, we can clearly see that 4 multiplied by 10 will give a result positive.
Let’s try for x = 10 => 4 x 10 – 38 = 40 – 38 = 2.
Hence answer is 10.

Example 2: Simplify (a + b) (3a – 3b + c) – (3a – 3b) c

Solution 2: Let’s simplify two parts of equation first and then we’ll combine to find final solution:
=> (a + b)(3a – 3b +c) and (3a – 3b)c
=> a(3a – 3b +c) + b(3a – 3b +c) and (3ac – 3bc)
=> 3a² – 3ab + ca + 3ab – 3b² + bc and 3ac – 3 bc
Now we’ll combine both equations, as shown below:
=> 3a² – 3ab + ca + 3ab – 3b² + bc – 3ac + 3 bc
=> 3a² – 3ab + 3ab + ca – 3ac – 3b² + bc + 3 bc
=> 3a² + 4bc – 2 ca – 3b²
=> 3(a² – b²) + 2c(2b – a)

Example 3: Multiply (a² + 4b²) and (7a – 4b)

Solution 3: a² (7a – 4b) + 4b² (7a – 4b)
=> 7a³ – 4a²b + 28ab² – 16b³
=> 7a³ – 16b³ – 4ab(a – 7b)

Example 4: Find value of (104)²

Solution 4: Let’s solve with the use of formula (x + y)² = (100 + 4)²
=> 100² + 4² + 2 x 100 x 4
=> 10000 + 16 + 800
=> 10816

Example 5: How many numbers less than 10 will give in a positive result for expression: a³ – 9a² + 10a +30?

Solution 5: Let’s assume a number between 1 to 10 to check final result is positive or not. If a = 6 then,
=> (6)³ – 9(6)² + 10 x 6 +30
=> 216 – 36 x 9 + 60 + 30 {Note: apply BODMAS}
=> 216 – 324 + 90
=> 306 – 324
=> -18

Now we’ll check for 7 number:
=> (7)³ – 9(7)² + 10 x 7 +30
=> 343 – 49 x 9 + 70 + 30 {Note: apply BODMAS}
=> 343 – 441 + 100
=> 443 – 441
=> 2
Hence numbers greater than 6 will give positive results.

Practical Questions on Basics of Algebra

Below are the few Practical Questions on the Basics of Algebra:

1. Find the value of x in expression 7x – 89, which will result in a positive number.
2. Find the value of x in expression 14x – 187, which will result in a positive number.
3. Simplify (2a – 3b) (3a + 3b + c) – (3a – 3b) c
4. Simplify (2a – 3b) (3a – 4b + c) + (8a – 3b) c
5. Multiply (a² – 3b²) and (7a + 4b)
6. Multiply (2a² + 9b²) and (5a + 8b)
7. Find the value of (106)²
8. Find the value of (56)²
9. How many numbers less than 10 will give in a positive result for expression: 2a³ – 10a² + 10a +30?
10. How many numbers less than 10 will give in a positive result for expression: 2a³ – 12a² + 13a +47?

Final Words

Practicing the above questions on Basics of Algebra is not the end of your practice, but it’s the start of a new journey to apply logic on Basics of Algebra with multiple approaches in a right and faster way.

For cracking competitive exams one must practice Basics of Algebra without a calculator is a must. At last, during the exam, if a solution for the Basics of Algebra question cannot be found easily then mark that question to revisit and move ahead instead of wasting time and energy.