Essential Algebra Formulas and Expressions List | Algebra PDF Guide

Algebra Formulas in Maths

Algebra is a fundamental branch of mathematics that utilizes variables, symbols, and equations to solve complex problems. Mastering algebra formulas is essential for success in fields like science, engineering, economics, and finance. Algebra is a core subject throughout school and university studies, and it remains a critical component of competitive exams such as the SSC, Banking, and RRB. Developing a strong foundation in algebraic expressions and formulas is the key to unlocking advanced mathematical proficiency and career success. 

In this comprehensive guide, we explore the most essential algebra formulas and expressions frequently applied in mathematics and real-world scenarios.

Algebraic Expressions Formula

An Algebraic Expression is a mathematical phrase comprised of variables, constants, and operators such as addition, subtraction, and multiplication. There are three primary types of algebraic expressions, which we define below.

Monomial Expression:

A monomial expression contains only a single term, such as 2x or 6y.

Binomial Expression:

A binomial expression consists of two terms, such as 6xy+5 or xy+y².

Polynomial Expression:

A polynomial expression features more than two terms with non-negative integral exponents, such as 6x²+4x+7 or 3y³+5y+15. 

Basic Algebra Formulas 

Algebraic equations are formed by combining numbers, variables, and operators. In these expressions, constants have known values, while variables represent unknown quantities. We use algebra formulas to systematically solve for these unknowns. The basic algebra formulas are summarized in the table below.

Basic Algebra Formulas

• (a+b)² = a² + 2ab + b²
• (a-b)² = a² – 2ab + b²
• a² – b² = (a-b)(a+b)
• a² + b² = (a-b)² +2ab
• (a+b+c)² = a²+b²+c²+2ab+2ac+2bc
• (a-b-c)² = a²+b²+c²-2ab-2ac+2bc
• (a+b)³ = a³+ 3a²b + 3ab² + b³
• (a-b)³ = a³- b³ + 3ab² - 3a²b
• a³-b³ = (a² + ab + b²)(a - b)
• a³+b³ = (a² – ab + b²)(a + b)
• (a+b) (a-b) = a2 – b2
• (x + a)(x + b) = x2 + (a + b)x + ab
• (x + a)(x – b) = x2 + (a – b)x – ab
• (x – a)(x + b) = x2 + (b – a)x – ab
• (x – a)(x – b) = x2 – (a + b)x + ab
• (a + b)3 = a3 + b3 + 3ab(a + b)
• (a – b)3 = a3 – b3 – 3ab(a – b)
• (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
• (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
• (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
• (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
• x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz -xz)
• x2 + y2 =½ [(x + y)2 + (x – y)2]
• (x + a) (x + b) (x + c) = x3 + (a + b +c)x2 + (ab + bc + ca)x + abc
• x3 + y3= (x + y) (x2 – xy + y2)
• x3 – y3 = (x – y) (x2 + xy + y2)
• x2 + y2 + z2 -xy – yz – zx = ½ [(x-y)2 + (y-z)2 + (z-x)2]

Algebra Formula Chart

Algebra is a broad mathematical field encompassing number theory, analysis, and geometry. From elementary equations to abstract structures, these concepts are vital for competitive exams. The complete list of core algebra formulas and expressions is provided below.

Algebra Formulas Chart
1.	a⁴ – b⁴ = (a² + b²) (a² – b²)
2.	a⁵ – b⁵ = (a – b)(a⁴+ a³b + a²b² + ab³ + b⁴ )
3.	a⁵ + b⁵ = (a + b)(a⁴  – a³b + a²b²– ab³ + b⁴ )
4.	a³ + b³+ c³– 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)
5.	(a + b + c+...)² = a²+b²+c²+...+2(ab + bc+....)
6.	If n is a natural number, a^n − b^n = (a−b)(a^(n−1) + a^(n−2) b+...+b^(n−2) a + b^(n−1))
7.	If n is even (n=2k), a^n + b^n = (a+b)(a^(n−1) − a^(n−2) b+...+b^(n−2) a − b^(n−1))
8.	If n is odd (n=2k+1), a^n + b^n = (a+b)(a^(n−1) − a^(n−2) b +...−b^(n−2) a + b^(n−1))
9.	(x+y+z)²=x²+y²+z²+2xy+2yz+2xz
10.	(x+y−z)²=x²+y²+z²+2xy−2yz−2xz
11.	(x−y+z)²=x²+y²+z²−2xy−2yz+2xz
12.	(x−y−z)²=x²+y²+z²−2xy+2yz−2xz
13.	x³+y³+z³−3xyz=(x+y+z)(x²+y²+z²−xy−yz−xz)
14.	(x+a)(x+b)(x+c)=x³+(a+b+c)x²+(ab+bc+ca)x+abc
15.	x²+y²+z²−xy−yz−zx=1/2[(x−y)²+(y−z)²+(z−x)²]

1. Quadratic Formula: The quadratic formula is used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.

The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a

This formula solves for x in a quadratic equation. Note that if the discriminant (the value under the square root) is negative, the equation yields no real number solutions.

1. Distance Formula: The distance formula is used to find the distance between two points in a coordinate plane. If the coordinates of the two points are (x1, y1) and (x2, y2), then the distance formula is as follows:

Distance formula: d = √((x2 - x1)² + (y2 - y1)²)

This formula is derived from the Pythagorean theorem and can be extended to three-dimensional space by including a z-coordinate.

1. Slope Formula: The slope formula is used to find the slope of a line passing through two points (x1, y1) and (x2, y2). The formula is as follows:

Slope formula: m = (y2 - y1) / (x2 - x1)

The slope (m) indicates the rate of change of y with respect to x on a coordinate plane.

1. Exponential Formula: The exponential formula is used to represent exponential growth or decay. If a quantity A grows exponentially over time t at a rate of r, then the formula is as follows:

Exponential growth formula: A = A0 e^(rt)

Here, A0 represents the initial quantity and e is the base of the natural logarithm.

For quantities that decay exponentially over time (t) at a rate (r), the formula is:

Exponential decay formula: A = A0 e^(-rt)

Where A0 is the initial quantity before the decay process begins.

1. Logarithmic Formula: The logarithmic formula is used to represent the inverse of exponential growth or decay. If y = b^x, then the logarithmic formula is as follows:

Logarithmic form: logb(y) = x

This relates to the exponent x where y = b^x. While any positive base is valid, base 10 and Euler's number (e) are the most common.

1. Pythagorean Theorem: The Pythagorean theorem is used to find the length of the sides of a right triangle. If a and b are the lengths of the two legs of a right triangle and c is the length of the hypotenuse, then the formula is as follows:

Pythagorean theorem: c² = a² + b²

This essential geometry formula allows you to find the third side of a right triangle when two sides are known.

1. Factorial Formula: The factorial formula is used to find the factorial of a number. The factorial of a positive integer n is the product of all positive integers from 1 to n. The formula is as follows:

Factorial notation: n! = 1 * 2 * 3 * ... * n

For example, 5! = 1 * 2 * 3 * 4 * 5 = 120.

Mastering these fundamental formulas will empower you to solve a wide array of algebraic problems across various mathematical disciplines.

 

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Laws of Exponents:

Understanding the basic laws of exponents is crucial for simplifying complex expressions. These laws allow students to compute high-value exponential terms efficiently without tedious expansion. 

• (a^m)(a^n)=a^(m+n)
• (ab)^m=a^m b^m
• (a^m)^n=a^(mn)

Fractional Exponents:

• a^0=1
• a^m / a^n = a^ (m−n)
• a^m = 1/a^(−m)
• a^(−m) = 1/a^m

Algebra Formulas Related Questions

Practice these essential algebra questions to sharpen your skills and deepen your conceptual understanding. 

Question 1: Evaluate the expression 8y + 6 when y = 3.

Solution: Substituting the value of y = 3 into the expression:

8(3) + 6 = 30  

Question 2: A class has 47 boys. The number of boys is three more than four times the number of girls. How many girls are in the class?

Solution: Let y represent the number of girls.

Formulating the equation based on the problem statement:

Number of boys = 3 + 4y = 47

4y = 44

Solving for y: y = 44 / 4 = 11. There are 11 girls in the class.

Question 3: Solve for y in the equation (y - 1)² = [4√(y - 4)]²

Solution: y² - 2y + 1 = 16(y - 4)

y² - 2y + 1 = 16y - 64

y² - 18y + 65 = 0

(y - 13)(y - 5) = 0

Thus, the solutions are y = 13 or y = 5.