Last updated on July 4th, 2025
In mathematics, algebra formulas are important as they form the foundation for polynomials, calculus, trigonometry, and quadratic equations. These formulas help solve and simplify algebraic expressions. In this article, algebraic formulas will be discussed in detail.
Algebra formulas are rules or equations that help with factoring, expanding, and simplifying expressions. We can use these formulas to solve complex algebraic equations efficiently. Here are some algebraic formulas:
To solve the expressions involving powers or exponents, we use the exponent rules. These rules are used to simplify expressions with powers. The exponent rules are:
Rule | Formula |
Product Rule | am × an = am + n |
Quotient Rule | am ÷ an = am - n |
Power of a Power Rule | (am)n = amn |
Power of a Product Rule | (ab)m = ambm |
Power of a Quotient Rule | (a/b)m = (am)/(bm) |
Zero Exponent Rule | a0 = 1 |
Negative Exponent Rule | a-m = 1/am |
Logarithms are used to solve multiplication and division of numbers with powers in simple ways. This makes them an effective tool to work with algebraic formulas with exponents. The relationship between the exponent and logarithm is: xm = a ⇒ logx a = m
Some commonly used log algebraic formulas are:
loga a = 1
loga 1 = 0
loga (xy) = loga x+ loga y
loga (x/y) = loga x - loga y
loga (xm) = m loga x
loga x = logcx/logca
a logax = x
The quadratic formula is one of the two methods to solve a quadratic equation. The standard form of a quadratic equation is ax2 + bx + c = 0. The value of the variable x can be found by using the formula:
x = -b ± √b2 - 4ac/2a
In algebra, permutations and combinations are formulas that help us identify the number of ways something can be arranged. Permutations refer to arrangements of items where the order matters, and combinations are the selection of items where order does not matter.
Factorial formula: n! = n × (n - 1) × (n - 2) × …. × 3 × 2 × 1
Permutations formula: nPr = n! / (n - r)!
Combination formula: nCr = n! / (r! (n - r)!)
Binomial theorem: (x + y)n = nC0xny0 + nC1xn-1y1 + nC2xn-2y2 + …. + nCn-1xyn-1 + nCnx0yn
The vector algebra formula is used to solve problems related to directions and magnitude. Some important vector formulas are:
For any three vectors a, b, and c in a 3D space
Algebraic identities are the equations that hold true for all values of the variables involved. It means LHS = RHS of the equation. Some common algebraic identities are -
The algebraic function expresses a relationship between two variables; it is written in the form y = f(x). Where x is the input and y is the output. For example, if x = 4, then f(x) = f(4) = 42 = 16.
In algebra, fractions that contain variables are called rational expressions. We can add, subtract, multiply, and divide fractions. The algebraic expression of fractions is:
In real life, we use algebraic formulas from managing personal finances and cooking to understanding scientific concepts and designing technology. Here are a few applications of algebraic formulas:
Students often think that algebra formulas are difficult and make mistakes. Here are some mistakes that students make and repeat, but by learning these mistakes, students can master algebra formulas.
Using algebra formulas, find (x +7)²?
(x + 7)2 = x2 + 14x + 49
To find the value of (x + 7)², we use the algebraic identity (a + b)2 = a2 + 2ab + b2
Here, a = x and b = 7
So, (x + 7)² = x² + (2 × x × 7) + 7²
= x² + 14x + 49
Find the value of (x + 2)(x + 8), using algebraic identity.
(x + 2) (x + 8) = x² + 10x + 16
The value of (x + 2) (x + 8) is calculated by using the identity (x + a) (x + b) = x2 + x(a + b) + ab
Here, a = 2 and b = 8
So, (x + 2) (x + 8) = x2 + x(a + b) + ab
= x² + x(2 + 8) + (2 × 8)
= x² + 10x + 16
Apply the identity (a + b) (a - b) = a² - b² to evaluate 102² - 98²
The value of 102² - 98² = 800
The value of 102² - 98² is calculated using the identity:
(a + b) (a - b) = a² - b², where a = 102 and b = 98
So, 1022 - 982 = (102 + 98) (102 - 98)
= 200 × 4 = 800
Solve the quadratic equation x² - 7x + 12 = 0 using quadratic formula
The value of x = 3 or x = 4
The quadratic formula is: x = -b ± √b2 - 4ac/2a
For x2 + bx + c = 0, here a = 1, b = -7, c = 12
x = -(-7) ± √(-7)2 - 4 × 1 × 12/2 × 1
= 7 ± √49 - 48/2 = (7 ± 1) / 2
So, x = (7 + 1)/2 = 8/2 = 4
x = (7 - 1)/2 = 6/2 = 3
Therefore, the value of x can be 4 or 3
Find the product of (x - 3)(x + 3) using algebraic formulas.
The value of (x - 3)(x + 3) = x2 -9
The value of (x - 3) (x + 3) is calculated by using the identity (a - b) (a + b) = a2 - b²
Here, a = x and b = 3
(a - b) (a + b) = x² - 3²
= x² - 9
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.