Podcast
Questions and Answers
Which of the following is the expansion of $(a + b)^2$?
Which of the following is the expansion of $(a + b)^2$?
What is the result of $(a - b)^2$?
What is the result of $(a - b)^2$?
How can $a^4 + a^2b^2 + b^4$ be factored?
How can $a^4 + a^2b^2 + b^4$ be factored?
Which of the following is equivalent to $a^3 - b^3$?
Which of the following is equivalent to $a^3 - b^3$?
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What is the formula for $a^2 - b^2$?
What is the formula for $a^2 - b^2$?
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गुरू तेग बहादुराने १६६५ साली कोणते शहर स्थापन केले?
गुरू तेग बहादुराने १६६५ साली कोणते शहर स्थापन केले?
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खालसा बंधूचे संस्थापक कोण होते?
खालसा बंधूचे संस्थापक कोण होते?
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गुरू तेग बहादुराचे मृत्यू केव्हा झाले?
गुरू तेग बहादुराचे मृत्यू केव्हा झाले?
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मीरी पीरी हा सिद्धांत काय दर्शवतो?
मीरी पीरी हा सिद्धांत काय दर्शवतो?
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गुरू तेग बहादुराने स्थापन केलेले शहर कोणत्या नदीच्या किनारी आहे?
गुरू तेग बहादुराने स्थापन केलेले शहर कोणत्या नदीच्या किनारी आहे?
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गुरू तेग बहादुराचा मृत्यू कोणत्या जागी झाला?
गुरू तेग बहादुराचा मृत्यू कोणत्या जागी झाला?
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Study Notes
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
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Description
Recognize and apply important algebraic formulas to solve equations and problems. This quiz covers formulas for squares, cubes, and higher powers of binomials.
