Algebraic Formulas

In algebra, a formula is an equation that shows the relationship between different variables. Variables represent unknown values that can change, while formulas show how those values are related mathematically. Formulas are important in algebra because they allow us to solve for an unknown variable by plugging in values we know for the other variables.

In this article, we will discuss list of algebraic formulas.

For quick and accurate calculations, try our Sum Of Squares Calculator.

Formulas in Algebra

Here is a list of important algebraic formulas:

• Difference of Squares: a² – b² = (a – b)(a + b)
• Square of a Binomial (Sum): (a + b)² = a² + 2ab + b²
• Sum of Squares: a² + b² = (a + b)² – 2ab
• Square of a Binomial (Difference): (a – b)² = a² – 2ab + b²
• Square of a Trinomial (Sum): (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• Square of a Trinomial (Difference): (a – b – c)² = a² + b² + c² – 2ab – 2bc – 2ca
• Cube of a Binomial (Sum): (a + b)³ = a³ + 3a²b + 3ab² + b³ ; (a + b)³ = a³ + b³ + 3ab(a + b)
• Cube of a Binomial (Difference): (a – b)³ = a³ – 3a²b + 3ab² – b³ = a³ – b³ – 3ab(a – b)
• Difference of Cubes: a³ – b³ = (a – b)(a² + ab + b²)
• Sum of Cubes: a³ + b³ = (a + b)(a² – ab + b²)
• Fourth Power of a Binomial (Sum): (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
• Fourth Power of a Binomial (Difference): (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴
• Difference of Fourth Powers: a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
• Fifth Power Difference: a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)
• General Formula for Powers with Natural Numbers: a_n – b_n = (a – b)(aⁿ⁻¹ + aⁿ⁻²b + … + abⁿ⁻² + bⁿ⁻¹)
• Sum and Difference of Even Powers: For even ‘n,’ a_n + b_n = (a + b)(aⁿ⁻¹ – aⁿ⁻²b + … – abⁿ⁻² – bⁿ⁻¹)
• Sum and Difference of Odd Powers: For odd ‘n,’ a_n+b_n = (a + b)(aⁿ⁻¹ – aⁿ⁻²b + aⁿ⁻³b² – … – abⁿ⁻² – bⁿ⁻¹)
• Sum of Squares of Multiple Numbers: (a + b + c + …)² = a² + b² + c² + … + 2(ab + ac + bc + …)
• Laws of Exponents: (a_m)(a_n) = a_m+n ; (ab)^m = a^mb^m ; (a^m)ⁿ =a^mn
• Zero Exponent Rule: a⁰ = 1

• Roots of Quadratic Equation: For ax² + bx + c = 0 (where ‘a’ ≠ 0), the roots are given by:

• Discriminant (Δ) = b² – 4ac
• For real and distinct roots, Δ > 0
• For real and coincident roots, Δ = 0
• For non-real roots, Δ < 0
• Sum of roots: α + β = -b/a
• Product of roots: α × β = c/a
• Quadratic equation in terms of roots: (x – α)(x – β) = 0

• Factorials: n! = 1 * 2 * 3 * … * (n – 1) * n ; 0! = 1

These formulas are essential in algebra and are frequently used to simplify expressions, solve equations, and understand mathematical relationships.

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