Addition of Squares

These problems involve the form $(a + b)^2 + (a – b)^2$:

1. $(3x + 5y)^2 + (3x – 5y)^2$

2. $(7a + 2b)^2 + (7a – 2b)^2$

3. $(x + \frac{4}{x})^2 + (x – \frac{4}{x})^2$

4. $(9m + n)^2 + (9m – n)^2$

5. $(2m + 7n)^2 + (2m – 7n)^2$

Subtraction of Squares

These problems involve the form $(a + b)^2 – (a – b)^2$:

1. $(4x + 3y)^2 – (4x – 3y)^2$

2. $(6a + b)^2 – (6a – b)^2$

3. $(p + \frac{5}{p})^2 – (p – \frac{5}{p})^2$

4. $(10x + y)^2 – (10x – y)^2$

5. $(x + \frac{1}{x})^2 – (x – \frac{1}{x})^2$

The Core Identities

• For Addition: $(a + b)^2 + (a – b)^2 = 2(a^2 + b^2)$

• For Subtraction: $(a + b)^2 – (a – b)^2 = 4ab$

1. Problems using $(a + b)^2 + (a – b)^2 = 2(a^2 + b^2)$

1. $(3x + 5y)^2 + (3x – 5y)^2$

   • Explanation: Here $a = 3x$ and $b = 5y$. Applying the identity, we get $2((3x)^2 + (5y)^2)$, which simplifies to $2(9x^2 + 25y^2) = \mathbf{18x^2 + 50y^2}$.

2. $(7a + 2b)^2 + (7a – 2b)^2$

   • Explanation: With $a = 7a$ and $b = 2b$, the result is $2((7a)^2 + (2b)^2)$. This becomes $2(49a^2 + 4b^2) = \mathbf{98a^2 + 8b^2}$.

3. $(x + \frac{4}{x})^2 + (x – \frac{4}{x})^2$

   • Explanation: Here $a = x$ and $b = \frac{4}{x}$. The identity gives $2(x^2 + (\frac{4}{x})^2)$, which simplifies to $2(x^2 + \frac{16}{x^2}) = \mathbf{2x^2 + \frac{32}{x^2}}$.

4. $(9m + n)^2 + (9m – n)^2$

   • Explanation: Substituting $a = 9m$ and $b = n$, we get $2((9m)^2 + n^2) = 2(81m^2 + n^2) = \mathbf{162m^2 + 2n^2}$.

5. $(2m + 7n)^2 + (2m – 7n)^2$

   • Explanation: Using $a = 2m$ and $b = 7n$, the identity results in $2((2m)^2 + (7n)^2) = 2(4m^2 + 49n^2) = \mathbf{8m^2 + 98n^2}$.

2. Problems using $(a + b)^2 – (a – b)^2 = 4ab$

6. $(4x + 3y)^2 – (4x – 3y)^2$

   • Explanation: Here $a = 4x$ and $b = 3y$. Using the subtraction identity $4ab$, we get $4(4x)(3y) = \mathbf{48xy}$.

7. $(6a + b)^2 – (6a – b)^2$

   • Explanation: With $a = 6a$ and $b = b$, the simplified form is $4(6a)(b) = \mathbf{24ab}$.

8. $(p + \frac{5}{p})^2 – (p – \frac{5}{p})^2$

   • Explanation: Here $a = p$ and $b = \frac{5}{p}$. Applying $4ab$ gives $4(p)(\frac{5}{p})$. The $p$ terms cancel out, leaving $4 \times 5 = \mathbf{20}$.

9. $(10x + y)^2 – (10x – y)^2$

   • Explanation: Substituting $a = 10x$ and $b = y$, we calculate $4(10x)(y) = \mathbf{40xy}$.

10. $(x + \frac{1}{x})^2 – (x – \frac{1}{x})^2$

    • Explanation: Here $a = x$ and $b = \frac{1}{x}$. Applying $4ab$ gives $4(x)(\frac{1}{x})$. The $x$ terms cancel, leaving $\mathbf{4}$.