Expand: $(3x + 4y)^2$
Expand: $(5a – 2b)^2$
Expand: $(7m + \frac{1}{n})^2$
Expand: $(4p – \frac{3}{q})^2$
For these, the document specifies to use a “suitable identity” to find the product: 5. $(6x + \frac{5}{x}) (6x + \frac{5}{x})$ 6. $(9y – \frac{2}{y}) (9y – \frac{2}{y})$
Use algebraic identities to evaluate the following numerical squares: 7. $103^2$ 8. $997^2$ 9. $204^2$ 10. $1005^2$
Identity I: $(a + b)^2 = a^2 + 2ab + b^2$
Identity II: $(a – b)^2 = a^2 – 2ab + b^2$
1. $(3x + 4y)^2$
Explanation: Use Identity I where $a = 3x$ and $b = 4y$.
Step: $(3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2$.
2. $(5a – 2b)^2$
Explanation: Use Identity II where $a = 5a$ and $b = 2b$.
Step: $(5a)^2 – 2(5a)(2b) + (2b)^2 = 25a^2 – 20ab + 4b^2$.
3. $(7m + \frac{1}{n})^2$
Explanation: Use Identity I with a fraction. Here, $a = 7m$ and $b = \frac{1}{n}$.
Step: $(7m)^2 + 2(7m)(\frac{1}{n}) + (\frac{1}{n})^2 = 49m^2 + \frac{14m}{n} + \frac{1}{n^2}$.
4. $(4p – \frac{3}{q})^2$
Explanation: Use Identity II where $a = 4p$ and $b = \frac{3}{q}$.
Step: $(4p)^2 – 2(4p)(\frac{3}{q}) + (\frac{3}{q})^2 = 16p^2 – \frac{24p}{q} + \frac{9}{q^2}$.
5. $(6x + \frac{5}{x}) (6x + \frac{5}{x})$
Explanation: Since both terms are identical, this is the same as $(6x + \frac{5}{x})^2$. Use Identity I.
Step: $(6x)^2 + 2(6x)(\frac{5}{x}) + (\frac{5}{x})^2$. Note that the $x$ variables in the middle term cancel out ($x \cdot \frac{1}{x} = 1$), leaving $36x^2 + 60 + \frac{25}{x^2}$.
6. $(9y – \frac{2}{y}) (9y – \frac{2}{y})$
Explanation: This is $(9y – \frac{2}{y})^2$. Use Identity II.
Step: $(9y)^2 – 2(9y)(\frac{2}{y}) + (\frac{2}{y})^2 = 81y^2 – 36 + \frac{4}{y^2}$.
For these, we rewrite the numbers as a sum or difference of “easy” numbers (like multiples of 10 or 100).
7. $103^2$
Explanation: Write as $(100 + 3)^2$ and use Identity I.
Step: $100^2 + 2(100)(3) + 3^2 = 10,000 + 600 + 9 = 10,609$.
8. $997^2$
Explanation: Write as $(1000 – 3)^2$ and use Identity II.
Step: $1000^2 – 2(1000)(3) + 3^2 = 1,000,000 – 6,000 + 9 = 994,009$.
9. $204^2$
Explanation: Write as $(200 + 4)^2$ and use Identity I.
Step: $200^2 + 2(200)(4) + 4^2 = 40,000 + 1,600 + 16 = 41,616$.
10. $1005^2$
Explanation: Write as $(1000 + 5)^2$ and use Identity I.
Step: $1000^2 + 2(1000)(5) + 5^2 = 1,000,000 + 10,000 + 25 = 1,010,025$.