Algebraic Formulas

Algebraic Formulas

Squares and Differences

• $(a + b)^2 = a^2 + b^2 + 2ab$
• $(a - b)^2 = a^2 + b^2 - 2ab$
• $a^2 - b^2 = (a + b)(a - b)$

Sums and Differences of Cubes

• $a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (a + b)^3 - 3ab(a + b)$
• $a^3 - b^3 = (a - b)(a^2 + ab + b^2) = (a - b)^3 + 3ab(a - b)$

Sum and Difference of Squares and Cubes

• $2(a^2 + b^2) = (a + b)^2 + (a - b)^2$
• $(a + b)^3 - (a - b)^3 = 4ab$
• $a^4 + b^4 = (a + b)(a - b)[(a + b)^2 - 2ab]$

Square of Sums and Differences

• $(a + b)^2 = (a - b)^2 + 4ab$
• $(a - b)^2 = (a + b)^2 - 4ab$
• $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
• $(a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2bc - 2ca$
• $(a - b - c)^2 = a^2 + b^2 + c^2 - 2ab + 2bc - 2ca$

Cubic Sum and Difference

• $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

Fourth Powers and Products

• $a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$
• $a^4 + a^2 + 1 = (a^2 + a + 1)(a^2 - a + 1)$
• $a^8 - b^8 = (a^4 + b^4)(a^2 + b^2)(a + b)(a - b)$

Special Cases

• If $a + b + c = 0$ then $a^3 + b^3 + c^3 = 3abc$

Guru Tegh Bahadur

Life and Contributions

• Founded the city of Anandpur Sahib in 1665, which became a center of Sikh learning and culture
• The city was located on the banks of the Sutlej River in the foothills of the Himalayas

Sikhism and Spirituality

• 9th Sikh Guru, spiritual leader, poet, and warrior
• Emphasized the importance of spiritual growth and selfless service to humanity

The Khalsa

• Conceptualized the Khalsa, a Sikh brotherhood that emphasizes courage, loyalty, and devotion to the Sikh faith
• Laid the foundation for the Khalsa, which was later formalized by Guru Gobind Singh

Martyrdom and Legacy

• Martyred in 1675 on the orders of Mughal Emperor Aurangzeb for refusing to convert to Islam
• Beheaded in Delhi's Chandni Chowk, symbolizing Sikh resistance against oppression and persecution

Miri Piri and Sikh Philosophy

• Embodied the concept of Miri Piri, combining spiritual and temporal power
• Believed spiritual growth and political power are interconnected and inseparable
• Reflected in the Sikh philosophy of combining spiritual devotion with worldly responsibility