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Lucía wants to design a square rug for her bedroom. The side of the rug measures $(x + 4)$ meters.

1. a) Write the expression for the area of the rug as the square of a binomial.
2. b) Expand the binomial to obtain the polynomial representing the area.

Expand the following expressions using algebraic identities to convert them into an ordered polynomial:

1. a) $(3x + 5)^2$

1. b) $(2x - 7y)^2$

1. c) $(x^2 + 4) \times (x^2 - 4)$

1. d) $(\frac{x}{2} - 6)^2$

Fill in the blanks so that the following equations are correct:

1. a) $(x + 4)^2 = x^2 + \dots x + 16$

1. b) $(a - \dots)^2 = a^2 - 6a + 9$

1. c) $(y + 5)(y - 5) = y^2 - \dots$

Copy and complete in your exercise book the missing terms so that these expressions form a perfect square trinomial or a difference of two squares:

1. a) $x^2 + 10x + \_\_\_\_ = (x + \_\_\_\_)^2$

1. b) $25a^2 - \_\_\_\_ + 1 = (\_\_\_\_ - 1)^2$

1. c) $16x^2 - \_\_\_\_ = (\_\_\_\_ + 3y) \times (\_\_\_\_ - 3y)$

Match each algebraic identity product in your exercise book with its corresponding expanded expression:

• A. $(x + 2)^2$

• B. $(x - 2)^2$

• C. $(x + 2)(x - 2)$

• $x^2 - 4$

• $x^2 + 4x + 4$

• $x^2 - 4x + 4$

Algebraic identities are also useful for performing quick mental calculations with large numbers. Use the difference of two squares identity (product of a sum and a difference) $(a + b) \times (a - b) = a^2 - b^2$ to calculate the following products cleverly:

1. a) $52 \times 48$ (Hint: consider $50 + 2$ and $50 - 2$)

1. b) $99 \times 101$

1. c) $35 \times 25$

A very common mistake when starting with algebraic identities is thinking that the square of a sum is simply the sum of the squares. For example:

Is it true that $(x + 3)^2 = x^2 + 3^2$?

To check this, calculate the numerical value of both expressions for $x = 2$ and explain whether the results are equal or different.

Lucía and Diego are doing their math homework and have made some common mistakes. Identify which of the following equations is incorrect and correct it in your notebook, explaining why it is wrong:

1. a) $(x + 4)^2 = x^2 + 16$

1. b) $(x - 5) \times (x + 5) = x^2 - 25$

1. c) $(2a - 3)^2 = 4a^2 - 12a + 9$

Simplify the following algebraic expression as much as possible by expanding using algebraic identities first and then collecting like terms:

$E = (2x + 1)^2 - (2x - 1)^2 + (x + 1) \times (x - 1)$

Look at the following figure representing a square with side length $(x + 3)$ from which an inner square of side length $(x - 1)$ has been cut out. Use algebraic identities to write a simplified expression representing the orange shaded area.

A local municipality wants to design a new square town square. In the center there will be a landscaped garden area, also square, surrounded by a paved path of constant width.

If the total side length of the square is $(3x + 2)$ meters and the side length of the central garden is $(3x - 2)$ meters, calculate the simplified polynomial that represents the area intended solely for the paved path.

Find the area of the shaded region in the following figure. Express the result as a simplified polynomial in terms of $x$. The side of the outer square measures $2x+3$ and the side of the inner white square measures $2x-3$.

Simplify the following algebraic fractions using algebraic identities to factorise the numerator and the denominator:

1. a) $\frac{x^2 - 25}{x^2 + 10x + 25}$

1. b) $\frac{9x^2 - 12x + 4}{9x^2 - 4}$