Important Algebra Formulas with Examples and Methods
Algebra uses letters and symbols to represent numbers and relationships. This chapter covers algebra basics, identities, indices, linear and quadratic equations, inequalities, factorisation and common algebra questions. The formulas and examples provide direct methods for simplifying expressions and finding unknown values in quantitative aptitude problems.
What Is Algebra?
An algebraic expression may contain constants, variables and coefficients. In 5x + 7, x is the variable, 5 is the coefficient and 7 is the constant. Terms with the same variables and powers are like terms, so 3x + 5x = 8x, while x and x² are unlike terms.
Algebra Video Lessons
Watch short topic-wise lessons for quick revision.
Square Identities in Algebra
Understand the identities (A+B)², (A−B)², and A²−B², including their expansions and factorisation forms for solving algebraic expressions.
Algebra Formula & Tricks
Important Formulas
The square of a binomial contains the squares of both terms and twice their product.
The middle term is negative because the two terms have opposite signs.
This identity factorises an expression formed by subtracting two perfect squares.
Add the square of each term and twice every pairwise product.
The formula gives both roots of a quadratic equation.
These rules apply when the base is the same for multiplication or division. For division, a ≠ 0.
Quick Tricks
If a quadratic expression can be split into two factors, factorisation is usually shorter than applying the quadratic formula.
Use (a + b)² or (a − b)² when a number is close to 10, 100 or another convenient value.
When the same expression occurs several times, assign it a temporary variable to reduce the calculation.
A term changes its sign when moved across the equality sign. Multiplying or dividing both sides by a negative number also reverses an inequality sign.
Algebra Concepts
Algebraic Expressions and Simplification
Like terms have identical variables raised to identical powers. The distributive property is a(b + c) = ab + ac. Constants can be combined with constants, but unlike terms cannot be added directly.
Algebraic Identities and Factorisation
Common identities include (a + b)², (a − b)² and a² − b². In factorisation, first take out the common factor, then use an identity or split the middle term when required.
Linear Equations in One Variable
Perform the same addition, subtraction, multiplication or division on both sides. For an equation ax + b = c with a ≠ 0, x = (c − b) ÷ a. Fractions can be cleared by multiplying every term by the least common denominator.
Quadratic Equations
The discriminant D = b² − 4ac determines the roots. D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives no real roots. The sum of roots is −b/a and their product is c/a.
Algebraic Inequalities
Solve an inequality like an equation while maintaining the correct order. For example, 2x + 3 ≤ nine? Actually use proper. 2x + 3 ≤ 11 gives 2x ≤ 8 and x ≤ 4. If −3x > 12, division by −3 reverses the sign, giving x < −4.
Algebra Data Table
Use the structured data below while following the explanation and examples.
Common Algebraic Identities
Frequently used identities for expansion and factorisation
| Identity type | Formula |
|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² |
| Square of a difference | (a − b)² = a² − 2ab + b² |
| Difference of squares | a² − b² = (a − b)(a + b) |
| Product of binomials | (x + a)(x + b) = x² + (a + b)x + ab |
Note: The variables may take any values for which the expressions are defined.
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Algebra Revision Points
Key formulas, rules and properties for quick revision
- Combine only like terms; unlike terms remain separate.
- Use a(b + c) = ab + ac to remove brackets.
- Remember (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
- Factorise a² − b² as (a − b)(a + b).
- For ax + b = c, x = (c − b)/a, where a ≠ 0.
- For ax² + bx + c = 0, use D = b² − 4ac to classify the roots.
- The sum and product of quadratic roots are −b/a and c/a respectively.
- Reverse an inequality sign when multiplying or dividing by a negative number.
Algebra FAQs
What is the value of (x + 3)²?
(x + 3)² = x² + 6x + 9, using (a + b)² = a² + 2ab + b².
How do you solve 2x + 7 = 19?
Subtract 7 from both sides: 2x = 12. Divide by 2, so x = 6.
What does the discriminant show in a quadratic equation?
For ax² + bx + c = 0, D = b² − 4ac. D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives no real roots.
What is the factorisation of x² − 16?
It is a difference of squares: x² − 16 = (x − 4)(x + 4).
What is the value of x² + 1/x² if x + 1/x = 6?
Square the given relation: x² + 2 + 1/x² = 36. Therefore, x² + 1/x² = 34, provided x ≠ 0.
When does an inequality sign change direction?
The sign changes when both sides are multiplied or divided by a negative number. For example, −2x > 10 gives x < −5.