1. The problem is to understand and explain the algebraic identities related to squares of sums and differences of variables $a$, $b$, and $c$. 2. The main formulas (वर्ग नियम सूत्रावलि) are: - $(a + b)^2 = a^2 + 2ab + b^2$ - $(a - b)^2 = a^2 - 2ab + b^2$ - $a^2 + b^2 = (a + b)^2 - 2ab$ - $a^2 + b^2 = (a - b)^2 + 2ab$ - $(a + b)^2 = (a - b)^2 + 4ab$ - $(a - b)^2 = (a + b)^2 - 4ab$ - $a^2 + b^2 = \frac{(a + b)^2 + (a - b)^2}{2}$ - $ab = \left(\frac{a + b}{2}\right)^2 - \left(\frac{a - b}{2}\right)^2$ - $4ab = (a + b)^2 - (a - b)^2$ - $a^2 - b^2 = (a + b)(a - b)$ (algebraic identity) - $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$ - $a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ac)$ - $2(ab + bc + ac) = (a + b + c)^2 - (a^2 + b^2 + c^2)$ 3. These formulas help in expanding and simplifying expressions involving squares of sums or differences, and in expressing sums of squares in terms of sums and products. 4. For example, to expand $(a + b)^2$, use formula 1: $(a + b)^2 = a^2 + 2ab + b^2$. 5. To find $a^2 + b^2$ using $(a + b)^2$ and $ab$, use formula 3(i): $a^2 + b^2 = (a + b)^2 - 2ab$. 6. The formula $a^2 - b^2 = (a + b)(a - b)$ is useful for factoring differences of squares. 7. For three variables, $(a + b + c)^2$ expands to $a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$. These identities are fundamental in algebra for simplifying expressions and solving equations.