Expanding Algebraic Expressions using Identities Worksheets - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. We will use algebraic identities to expand each expression.

1) $\left(\frac{1}{4}x + y^2\right)^2$
* Identity: $(a + b)^2 = a^2 + 2ab + b^2$
* Let $a = \frac{1}{4}x$ and $b = y^2$.
* $a^2 = (\frac{1}{4}x)^2 = \frac{1}{16}x^2$
* $2ab = 2 \cdot (\frac{1}{4}x) \cdot (y^2) = \frac{2}{4}xy^2 = \frac{1}{2}xy^2$
* $b^2 = (y^2)^2 = y^4$
* Result: $\frac{1}{16}x^2 + \frac{1}{2}xy^2 + y^4$

2) $(4.2p - 5q)^2$
* Identity: $(a - b)^2 = a^2 - 2ab + b^2$
* Let $a = 4.2p$ and $b = 5q$.
* $a^2 = (4.2p)^2 = 17.64p^2$
* $2ab = 2 \cdot (4.2p) \cdot (5q) = 42pq$
* $b^2 = (5q)^2 = 25q^2$
* Result: $17.64p^2 - 42pq + 25q^2$

3) $\left(1 - \frac{x}{5}\right)^2$
* Identity: $(a - b)^2 = a^2 - 2ab + b^2$
* Let $a = 1$ and $b = \frac{x}{5}$.
* $a^2 = 1^2 = 1$
* $2ab = 2 \cdot 1 \cdot \frac{x}{5} = \frac{2x}{5}$
* $b^2 = (\frac{x}{5})^2 = \frac{x^2}{25}$
* Result: $1 - \frac{2x}{5} + \frac{x^2}{25}$

4) $\left(z + \frac{2w}{7}\right)^2$
* Identity: $(a + b)^2 = a^2 + 2ab + b^2$
* Let $a = z$ and $b = \frac{2w}{7}$.
* $a^2 = z^2$
* $2ab = 2 \cdot z \cdot \frac{2w}{7} = \frac{4zw}{7}$
* $b^2 = (\frac{2w}{7})^2 = \frac{4w^2}{49}$
* Result: $z^2 + \frac{4zw}{7} + \frac{4w^2}{49}$

5) $(3.4 - 0.08r)^2$
* Identity: $(a - b)^2 = a^2 - 2ab + b^2$
* Let $a = 3.4$ and $b = 0.08r$.
* $a^2 = 3.4^2 = 11.56$
* $2ab = 2 \cdot 3.4 \cdot 0.08r = 6.8 \cdot 0.08r = 0.544r$
* $b^2 = (0.08r)^2 = 0.0064r^2$
* Result: $11.56 - 0.544r + 0.0064r^2$

6) $\left(-u - \frac{v}{3}\right)^2$
* Identity: $(a + b)^2 = a^2 + 2ab + b^2$ (Note: Since both terms are negative, squaring them makes them positive, so it acts like addition).
* Let $a = -u$ and $b = -\frac{v}{3}$.
* $a^2 = (-u)^2 = u^2$
* $2ab = 2 \cdot (-u) \cdot (-\frac{v}{3}) = \frac{2uv}{3}$
* $b^2 = (-\frac{v}{3})^2 = \frac{v^2}{9}$
* Result: $u^2 + \frac{2uv}{3} + \frac{v^2}{9}$

7) $(6p^2 + h^2)^2$
* Identity: $(a + b)^2 = a^2 + 2ab + b^2$
* Let $a = 6p^2$ and $b = h^2$.
* $a^2 = (6p^2)^2 = 36p^4$
* $2ab = 2 \cdot (6p^2) \cdot (h^2) = 12p^2h^2$
* $b^2 = (h^2)^2 = h^4$
* Result: $36p^4 + 12p^2h^2 + h^4$

8) $\left(\frac{k^2}{2} - 0.5\right)^2$
* Identity: $(a - b)^2 = a^2 - 2ab + b^2$
* Let $a = \frac{k^2}{2}$ and $b = 0.5$ (which is $\frac{1}{2}$).
* $a^2 = (\frac{k^2}{2})^2 = \frac{k^4}{4}$
* $2ab = 2 \cdot \frac{k^2}{2} \cdot 0.5 = k^2 \cdot 0.5 = 0.5k^2$ (or $\frac{1}{2}k^2$)
* $b^2 = (0.5)^2 = 0.25$ (or $\frac{1}{4}$)
* Result: $\frac{k^4}{4} - 0.5k^2 + 0.25$

9) $\left(\frac{m}{4} - \frac{n}{5}\right)^2$
* Identity: $(a - b)^2 = a^2 - 2ab + b^2$
* Let $a = \frac{m}{4}$ and $b = \frac{n}{5}$.
* $a^2 = (\frac{m}{4})^2 = \frac{m^2}{16}$
* $2ab = 2 \cdot \frac{m}{4} \cdot \frac{n}{5} = \frac{2mn}{20} = \frac{mn}{10}$
* $b^2 = (\frac{n}{5})^2 = \frac{n^2}{25}$
* Result: $\frac{m^2}{16} - \frac{mn}{10} + \frac{n^2}{25}$

10) $(8 + 3z)^2$
* Identity: $(a + b)^2 = a^2 + 2ab + b^2$
* Let $a = 8$ and $b = 3z$.
* $a^2 = 8^2 = 64$
* $2ab = 2 \cdot 8 \cdot 3z = 48z$
* $b^2 = (3z)^2 = 9z^2$
* Result: $64 + 48z + 9z^2$

Final Answer:
1) $\frac{1}{16}x^2 + \frac{1}{2}xy^2 + y^4$
2) $17.64p^2 - 42pq + 25q^2$
3) $1 - \frac{2x}{5} + \frac{x^2}{25}$
4) $z^2 + \frac{4zw}{7} + \frac{4w^2}{49}$
5) $11.56 - 0.544r + 0.0064r^2$
6) $u^2 + \frac{2uv}{3} + \frac{v^2}{9}$
7) $36p^4 + 12p^2h^2 + h^4$
8) $\frac{k^4}{4} - 0.5k^2 + 0.25$
9) $\frac{m^2}{16} - \frac{mn}{10} + \frac{n^2}{25}$
10) $64 + 48z + 9z^2$