Algebra is a foundational branch of mathematics that deals with symbols and the rules for manipulating those symbols. It plays a crucial role in solving problems in arithmetic, geometry, and higher-level mathematics. Whether you’re a school student or preparing for competitive exams, understanding and remembering key algebra formulas can save time and improve problem-solving efficiency. Here’s a breakdown of some of the most important algebra formulas you should know.
1. Algebraic Identities
Algebraic identities are equations that are true for all values of the variables involved. They simplify the process of expanding and factoring expressions.
a. (a + b)² = a² + 2ab + b²
This identity is used to expand a binomial squared.
b. (a – b)² = a² – 2ab + b²
It’s similar to the first, but for subtraction.
c. a² – b² = (a + b)(a – b)
Known as the difference of squares, this is useful in factorization.
d. (x + a)(x + b) = x² + (a + b)x + ab
Useful for multiplying two binomials.
2. Quadratic Equation Formula
The standard form of a quadratic equation is:
ax² + bx + c = 0
To find the roots of the equation, use the quadratic formula:
x = [-b ± √(b² – 4ac)] / (2a)
Here, b² – 4ac is called the discriminant. If:
- It’s positive: two real roots
- Zero: one real root
- Negative: no real root (complex roots)
3. Factorization Formulas
These help in breaking down expressions into simpler multiplicative parts.
a. x² + (a + b)x + ab = (x + a)(x + b)
A common formula used in solving quadratic equations.
b. x³ + y³ = (x + y)(x² – xy + y²)
Sum of cubes identity.
c. x³ – y³ = (x – y)(x² + xy + y²)
Difference of cubes identity.
4. Laws of Exponents
Exponents appear frequently in algebra, and knowing these laws helps simplify expressions:
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (where a ≠ 0)
- (ab)ⁿ = aⁿbⁿ
- (a/b)ⁿ = aⁿ / bⁿ
5. Linear Equations
A linear equation in one variable is in the form:
ax + b = 0
To solve it, isolate x:
x = -b/a
For two variables:
ax + by = c
Solving requires methods like substitution or elimination.
6. Arithmetic Progression (AP)
In algebra, progressions are sequences with a fixed pattern.
- nth term (Tₙ) = a + (n – 1)d
- Sum of n terms (Sₙ) = n/2 × [2a + (n – 1)d]
Where:
- a = first term,
- d = common difference,
- n = number of terms
Conclusion
These algebra formulas form the core of many mathematical applications. Memorizing them and practicing regularly will build your confidence and speed in solving algebra problems. Whether simplifying expressions or solving equations, these tools make the job quicker and easier. Always try to understand the derivations — it’ll make remembering them much simpler!
