Algebraic Expressions Formula

Algebra Formulas

Algebra is a branch of mathematics that substitutes letters for numbers. An algebraic equation depicts a scale, what is done on one side of the scale with a number is also done to either side of the scale. The numbers are constants. Algebra also includes real numbers, complex numbers, matrices, vectors and much more. X, Y, A, B are the most commonly used letters that represent algebraic problems and equations.

Important Formulas in Algebra

Here is a list of Algebraic formulas –

• a² – b² = (a – b)(a + b)
• (a + b)² = a² + 2ab + b²
• a² + b² = (a + b)² – 2ab
• (a – b)² = a² – 2ab + b²
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca
• (a + b)³ = a³ + 3a²b + 3ab² + b³ ; (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – 3a²b + 3ab² – b³ = a³ – b³ – 3ab(a – b)
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)
• (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
• (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴
• a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
• a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)
• If n is a natural number aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b+…+ b^n-2a + b^n-1)
• If n is even (n = 2k), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +…+ b^n-2a – b^n-1)
• If n is odd (n = 2k + 1), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +a^n-3b²…- b^n-2a + b^n-1)
• (a + b + c + …)² = a² + b² + c² + … + 2(ab + ac + bc + ….)
• Laws of Exponents (a^m)(aⁿ) = a^m+n ; (ab)^m = a^mb^m ; (a^m)ⁿ = a^mn
• Fractional Exponents a⁰ = 1 ;

• Roots of Quadratic Equation

• For a quadratic equation ax² + bx + c = 0 where a ≠ 0, the roots will be given by the equation as

• Δ = b² − 4ac is called the discriminant
• For real and distinct roots, Δ > 0
• For real and coincident roots, Δ = 0
• For non-real roots, Δ < 0
• If α and β are the two roots of the equation ax² + bx + c = 0 then, α + β = (-b / a) and α × β = (c / a).
• If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

• Factorials

• n! = (1).(2).(3)…..(n − 1).n
• n! = n(n − 1)! = n(n − 1)(n − 2)! = ….
• 0! = 1

Solved Examples

Example 1: Find out the value of 5² – 3²
Solution:
Using the formula a² – b² = (a – b)(a + b)
where a = 5 and b = 3
(a – b)(a + b)
= (5 – 3)(5 + 3)

Algebra: Expressions And Equations

Mathematics is known as the “Queen of Science”. Algebra is a special branch of mathematics that deals with numbers, shapes and letters. We use concepts of mathematics everywhere, every day, almost in every situation. So far, we have used lots of numbers, lots of shapes and figures. Here, we are going to use some letters in math. Yes, it is algebra; let’s learn more about it.

Expressions and Equations

Mathematics has been classified into many branches. Arithmetic is the branch that handles numbers and their operations. Arithmetic taught us addition, subtraction, multiplication and divisions of two or more numbers. Geometry is all about shapes and their construction using different tools like a compass, ruler and pencil. Algebra is another interesting branch where we express our daily life situations in numbers and letters (variables).

To understand algebra we need to know what an expression is and what an equation is.

Expression Definition

An expression or algebraic expression is any mathematical statement which consists of numbers, variables and an arithmetic operation between them. For example, 4m + 5 is an expression where 4m and 5 are the terms and m is the variable of the given expression separated by the arithmetic sign +.

Variable is anything that varies; it doesn’t have a fixed value. Generally, expression variables are represented by alphabetical letters like a, b, c, m, n, p, x, y, z, and so on. We can form a variety of expressions using a combination of different variables and numbers.

How to Simplify Algebraic Expressions?

The purpose of simplifying the algebraic expression is to find the simplified term of the given expression. To factorize or simplify the expression, we should first know how to combine the like terms, factor a number, order of operations. In combining like terms, the variables with the same degree are gathered together, and the constant terms are separated for the simplification process.

Expression Examples

Few examples of expressions are as follows:

• x + 5y – 10
• 2x + 1
• x + y

Equation Definition

The equation is an expression where two sides are connected through an equal sign (=). 2x + 1 = 9 is an equation, where 2x+1 is the left-hand side (LHS) and 9 is the expression’s right-hand side (RHS). The equal sign between LHS and RHS indicates that the value of LHS is equal to the RHS of the expression.

10x + 63 > 10, is not an equation. Here, the sign between LHS and RHS of the expression is not an equal sign. Hence, we can say every expression is not an equation. These kinds of expressions are called linear inequalities.

Equation Examples

Few examples of equations are:

• 8m+5=10n
• a+4b = 12c+3

Note: An equation is interchangeable i.e. the equation remains the same even if LHS and RHS interchange each other.

Difference between Algebraic Expression and Equation

How to Solve Algebraic Equations?

An algebraic equation contains two algebraic expressions separated by an equal sign (=) in between. The primary purpose of solving algebraic equations is to find the unknown variable in the given expression. While solving the equation, separate the variable terms on one side and constant terms on another side. The variable term can be isolated using the various arithmetic operations such as addition, subtraction, multiplication, division, and other operations like finding square roots, etc.

Algebraic Expression and Equation Problem

Question: Find the value of x in the given equation: 4x + 10 = 30

Solution:

Given Equation: 4x + 10 = 30

Separate the variable and the constant term:

Keep the variable term on the left-hand side, and move the constant term on the right-hand side.

So, the given equation is written as:

4x = 30-10 [Performing subtraction operation]

4x = 20

x = 20/4 [Performing division operation]

x = 5.

Therefore, the value of x is 5.

Alternative Method:

Given equation is:

4x + 10 = 30

Subtracting 10 from both sides of the equation, we get;

4x + 10 – 10 = 30 – 10

4x = 20

Dividing both sides of equation by 4, we get;

4x/4 = 20/4

x = 5

What is Algebraic Equations?

An algebraic equation is a mathematical statement that contains two equated algebraic expressions. The general form of an algebraic equation is P = 0 or P = Q, where P and Q are polynomials. Algebraic equations that contain only one variable are known as univariate equations and those which contain more than one variable are known as multivariate equations. An algebraic equation will always be balanced. This means that the right-hand side of the equation will be equal to the left-hand side.

Algebraic Equations Examples

x² – 5x = 3 is a univariate algebraic equation while y²x – 5z = 3x is an example of a multivariate algebraic equation.

Example 1: Solve the algebraic equation x + 3 = 2x
Solution: Taking the variable terms on one side of the equation and keeping the constant terms on the other side we get,
3 = 2x – x
3 = x
Answer: x = 3

Example 2: A total of 15 items can fit in a box. If the box contains 2 scales, 7 pencils, and 1 eraser then how many pens can fit in the box?
Solution: Converting this problem statement in the form of an algebraic equation we get,
2 scales + 7 pencils + 1 eraser + x pens = 15
2 + 7 + 1 + x = 15
Solving the L.H.S
10 + x = 15
x = 15 – 10
x = 5
Answer: 5 pens can fit in the box

Example 3: Find the roots of the quadratic equation x² + x – 6 = 0
Solution: Using the quadratic formula x = [-b ± √(b² – 4ac)]/2a.
a = 1, b = 1, c = – 6
x = [-1 ± √(1² – 4 · 1 · -6)] / (2 · 1)
x = [-1 ± √(25)] / 2
x = [-1 + 5] / 2,  [-1 – 5] / 2
x = 2, -3
Answer: The roots of the given algebraic equation are 2 and -3.

Types of Algebraic Equations

Algebraic equations can be classified into different types based on the degree of the equation. The degree can be defined as the highest exponent of a variable in an algebraic equation. Suppose there is an equation given by x⁴ + y³ = 3⁵ then the degree will be 4. In determining the degree, the exponent of the constant or coefficient is not considered. The number of roots of an algebraic equation depends on its degree. An algebraic equation where the degree equals 5 will have a maximum of 5 roots. The various types of algebraic equations are as follows:

Linear Algebraic Equations

A linear algebraic equation is one in which the degree of the polynomial is 1. The general form of a linear equation is given as a₁x₁+a₂x₂+…+a_nx_n = 0 where at least one coefficient is a non-zero number. These linear equations are used to represent and solve linear programming problems.

Example: 3x + 5 = 5 is a linear equation in one variable. y = 2x – 6 is a linear equation in two variables.

Quadratic Algebraic Equations

An equation where the degree of the polynomial is 2 is known as a quadratic algebraic equation. The general form of such an equation is ax² + bx + c = 0, where a is not equal to 0.

Example: 3x² + 2x – 6 = 0 is a quadratic algebraic equation. This type of equation will have a maximum of two solutions.

Cubic Algebraic Equations

An algebraic equation where the degree equals 3 will be classified as a cubic algebraic equation. ax³ + bx² + cx + d = 0 is the general form of a cubic algebraic equation (a ≠ 0).

Example: x³ + x² – x – 1 = 0. A cubic algebraic equation will have a maximum of three roots as the degree is 3.

Higher-Order Polynomial Algebraic Equations

Algebraic equations that have a degree greater than 3 are known as higher-order polynomial algebraic equations. Quartic (degree = 4), quintic (5), sextic (6), septic (7) equations all fall under the category of higher algebraic equations. Such equations might not be solvable using a finite number of operations.

Important Notes on Algebraic Equations:

• An algebraic equation is an equation where two algebraic expressions are joined together using an equal sign.
• Polynomial equations are algebra equations.
• Algebraic equations can be one-step, two-step, or multi-step equations.
• Algebra equations are classified as linear, quadratic, cubic, and higher-order equations based on the degree.

FAQs on Algebraic Equations

What are Algebraic Equations?

Algebraic equations are polynomial equations where two algebraic expressions are equated. Both sides of the equation must be balanced. The general form of an algebraic equation is P = 0.

What is an Example of Algebraic Equation?

An algebraic equation can be linear, quadratic, etc. Hence, an example of an algebraic equation can be 3x² – 6 = 0.

How Do You Solve Algebraic Equations?

There are many methods available to solve algebraic equations depending on the degree. Some techniques include applying simple algebraic operations, solving simultaneous equations, splitting the middle term, quadratic formula, long division, and so on.

What are Algebraic Expressions and Algebraic Equations?

Mathematical statements that consist of variables, coefficients, constants, and algebraic operations are known as algebraic expressions. When two algebraic expressions are equated together, they are known as algebraic equations.

How Do You Write Algebraic Equation?

We can convert real-life statement involving numbers and conditions into algebraic equation. For example, if the problem says, “the length of a rectangular field is 5 more than twice the width”, then it can be written as the algebraic equations l = 2w + 5, where ‘l’ and ‘w’ are the length and width of the rectangular field.

What are Linear Algebraic Equations?

An algebraic equation where the highest exponent of the variable term is 1 is a linear algebraic equation. In other words, algebraic equations with degree 1 will be linear. For example, 3y – 9 = 1

Are Quadratic Equations Algebraic Equations?

Yes, quadratic equations are algebraic equations. It consists of an algebraic expression of the second degree.

What are the Basic Formulas of Algebraic Equations?

Some of the basic formulas of algebraic equations are listed below:

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• Quadratic Formula: [-b ± √(b² – 4ac)]/2a
• Discriminant: b² – 4ac

What are the Rules for Algebraic Equations?

There are 5 basic rules for algebraic equations. These are as follows:

• Commutative Rule of Addition
• Commutative Rule of Multiplication
• Associative Rule of Addition
• Associative Rule of Multiplication
• Distributive Rule of Multiplication

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