# Algebra Formula

## Algebra Formulas - Basic Algebraic Formulas and Expression

Starting from computer science to engineering, the importance of algebra is manifold in nearly every career choice. So, learners must grasp every algebra formula to prepare themselves for calculations beyond basic math. Not only for learners in school but even candidates appearing for competitive exams should have a firm grasp of algebraic formulae to excel.

“Algebra can be your stepping stone of success, opening doors of opportunities and excellent discoveries.”

Vedantu brings a comprehensive algebra formulas list to aid students in learning its basic as well as advanced concepts effortlessly.

Do you know?

• Algebra was introduced by the Greeks back in the 3rd century.

• It was the Babylonians who created the algebraic equation and formulae we still use in the 21st century to solve diverse problems.

• Modern algebra was brought in by Rene Descartes in the 16th century.

Sounds a bit exciting?

Here’s what is involved in the study of algebra

The study of algebra revolves around in-depth learning of terms, concepts and formulae. The idea is simple. Here, mathematical symbols, known as variables, represent quantity without having any fixed value and these are manipulated to derive solutions.

A basic example:

Algebra asks questions like what is the value of x if x + 7 = 10? To get the result, you need to do another calculation, i.e. 10 – 7 = 3. So, the value of x is 3.

Once you learn the basics (elementary algebra), advanced levels gradually become easier.

Master algebra for its many perks!

Well, we understand how tedious studying these alphanumerical formulae can be. However, if you know the perks of mathematics formula algebra in real life, you may find it convincing enough to learn. Here you go!

1. As algebraic formulae are rules that work in every circumstance, these largely help in decision making.

2. Studying it enables you to think logically and resolve complex problems efficiently.

3. Many subjects such as engineering, physics, chemistry, etc. need algebra. So, mastering the concepts and formulae can significantly help you in higher studies.

4. Without algebra, the internet, Google, digital televisions, mobile phones and other advanced technology would not have existed. It signifies that learning maths algebra formula can prepare you for a career in the technology sector.

Above all, mathematics teachers and toppers also recommend studying the formulae as these can save a lot of time during the examination.

Various algebraic formulas & expression

• • ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$

• • ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$

• • ${a^2} + {b^2} = {\left( {a - b} \right)^2} + 2ab$

• • ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$

• • ${\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2ac + 2bc$

• • ${\left( {a - b - c} \right)^2} = {a^2} + {b^2} + {c^2} - 2ab - 2ac + 2bc$

• • ${\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3};{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$

• • ${\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}$

• • ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$

• • ${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$

• • ${\left( {a + b} \right)^4} = {a^4} + 4{a^3}b + 6{a^2}{b^2} + 4a{b^3} + {b^4}$

• • ${\left( {a - b} \right)^4} = {a^4} - 4{a^3}b + 6{a^2}{b^2} - 4a{b^3} + {b^4}$

• • ${a^4} - {b^4} = \left( {a - b} \right)\left( {a + b} \right)\left( {{a^2} + {b^2}} \right)$

• • ${a^5} - {b^5} = \left( {a - b} \right)\left( {{a^4} + {a^3}b + {a^2}{b^2} + a{b^3} + {b^4}} \right)$

• • ${\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy + 2yz + 2xz$

• • ${\left( {x + y - z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy - 2yz - 2xz$

• • ${\left( {x - y + z} \right)^2} = {x^2} + {y^2} + {z^2} - 2xy - 2yz + 2xz$

• • ${\left( {x - y - z} \right)^2} = {x^2} + {y^2} + {z^2} - 2xy + 2yz - 2xz$

• • ${x^3} + {y^3} + {z^3} - 3xyz = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - xy - yz - xz} \right)$

• • ${x^2} + {y^2} = \frac{1}{2}\left[ {{{\left( {x + y} \right)}^2} + {{\left( {x - y} \right)}^2}} \right]$

• • $\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + bc + ca} \right)x + abc$

• • ${x^3} + {y^3} = \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)$

• • ${x^3} - {y^3} = \left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)$

• • ${x^2} + {y^2} + {z^2} - xy - yz - zx = \frac{1}{2}\left[ {{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}} \right]$

• • If n is a natural number, ${a^n} - {b^n} = \left( {a - b} \right)\left( {{a^{n - 1}} + {a^{n - 2}}b + ... + {b^{n - 2}}a + {b^{n - 1}}} \right)$

• • If n is even $\left( {n = 2k} \right),{a^n} + {b^n} = \left( {a + b} \right)\left( {{a^{n - 1}} - {a^{n - 2}}b + ... + {b^{n - 2}}a - {b^{n - 1}}} \right)$

• • If n is odd $\left( {n = 2k + 1} \right),{a^n} + {b^n} = \left( {a + b} \right)\left( {{a^{n - 1}} - {a^{n - 2}}b + ... - {b^{n - 2}}a + {b^{n - 1}}} \right)$

• • ${\left( {a + b + c + ...} \right)^2} = {a^2} + {b^2} + {c^2} + ... + 2\left( {ab + bc + ....} \right)$

• • Laws of Exponents

•  $\begin{gathered} \left( {{a^m}} \right)\left( {{a^n}} \right) = {a^{m + n}} \hfill \\ {\left( {ab} \right)^m} = {a^m}{b^m} \hfill \\ {\left( {{a^m}} \right)^n} = {a^{mn}} \hfill \\ \end{gathered}$
• • Fractional Exponents

• $\begin{gathered} {a^0} = 1 \hfill \\ \frac{{{a^m}}}{{{a^n}}} = {a^{m - n}} \hfill \\ {a^m} = \frac{1}{{{a^{ - m}}}} \hfill \\ {a^{ - m}} = \frac{1}{{{a^m}}} \hfill \\ \end{gathered}$

Practice Problem:
Find value of (3 + 7)2
Sol: Using formula ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
$\begin{gathered} {\left( {3 + 7} \right)^2} = {3^2} + {7^2} + 2\left( 3 \right)\left( 7 \right) \hfill \\ \quad \quad \quad \, = 9 + 49 + 42 \hfill \\ \quad \quad \quad \, = 100 \hfill \\ \end{gathered}$

Practice Question
Find the value of (4 + 3 − 2)2