Exercises: Standard algebraic identities

In a 2nd year of ESO exam, Lucía solved the expansion of the expression $(2x - 3)^2$ by directly writing: $4x^2 - 9$. Her teacher pointed out that the answer is wrong because it contains two conceptual errors in that single step.

1. a) Identify the two errors made by Lucía (one regarding the sign of the last term and another regarding the missing terms in the expansion).
2. b) Apply the distributive property by multiplying $(2x - 3) \cdot (2x - 3)$ term by term to clearly explain to her where each element comes from, and write the final correct and simplified expression.

To cover the floor of a square stand at the Feria del Libro de Madrid, a tarpaulin with a side length of $39\text{ m}$ needs to be ordered.

1. a) Before calculating the exact area, make a mental estimate by rounding the side length to the nearest ten ($40\text{ m}$). Reason whether the actual area of the stand will be greater than or less than your estimate.
2. b) Express the number $39$ as the subtraction $(40 - 1)$ and calculate the exact area by applying the remarkable identity of the square of a difference, indicating the three terms resulting from the expansion.
3. c) Subtract the calculated exact area from your initial estimate and check whether the difference matches what you reasoned in section a).

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) $(x^2 + 4) \cdot (x^2 - 4)$

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

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.

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

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

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

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

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) $(a + b) \cdot (a - b) = a^2 - b^2$ to calculate the following products cleverly:

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

1. b) $99 \cdot 101$

1. c) $35 \cdot 25$

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) $16x^2 - \_\_\_\_ = (\_\_\_\_ + 3y) \cdot (\_\_\_\_ - 3y)$

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

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

A local council wants to design a new square town square. In the centre 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 $(3x + 2)$ metres and the side length of the central garden is $(3x - 2)$ metres, calculate the simplified polynomial that represents the area intended solely for the paved path.

In the cube in the figure, with edge length $x$, the face diagonal of one of its faces and the main body diagonal have been drawn. In addition, a regular hexagon is formed by cutting the cube with a plane $P$ that passes through the midpoints of its edges.

1. a) Express as a squared binomial the area of a face if the edge length were increased by $2\text{ cm}$.
2. b) If the area of the hexagon can be expressed as $\frac{3\sqrt{3}}{4} x^2$, prove using algebraic identities that if the edge length were $(x+y)$, the new area of the hexagon would be $\frac{3\sqrt{3}}{4}(x^2 + 2xy + y^2)$.
3. c) Calculate the exact perimetre of the hexagon for $x = 4\text{ cm}$ (simplify the square root).

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

• A. $(x + 2)^2$

• B. $(x - 2)^2$

• C. $(x + 2)(x - 2)$

• $x^2 - 4$

• $x^2 + 4x + 4$

• $x^2 - 4x + 4$

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

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

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.

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.

To mentally calculate the result of the operation $105^2 - 95^2$ without using a calculator, Marcos and Elena propose two different strategies:

• Marcos's method: Perform the two traditional multiplications separately, $105 \cdot 105$ and $95 \cdot 95$, and then subtract the second result from the first.

• Elena's method: Identify that the expression is a difference of squares of the form $a^2 - b^2$, transform it into the product of a sum and a difference, and perform the resulting operations.

1. a) Apply the method proposed by Elena, expanding the numerical calculation step by step until obtaining the final value.
2. b) Compare both methods and justify with reasons why Elena's proposal is much more advantageous, analyzing the difficulty of the intermediate operations for each path.

An irrigation community in the huerta de Murcia has a square crop plot with a side length of $x$ meters (with $x > 10$). To improve the irrigation network, two remodelling options are being studied:

• Option A: Increase one of the sides by $5\text{ m}$ and reduce the perpendicular side by $5\text{ m}$ to dig an irrigation ditch.

• Option B: Increase both perpendicular sides by $5\text{ m}$ to take advantage of an adjacent strip of land.

1. a) Write the formula for the resulting area in Option A as the product of two binomials, expand it using the corresponding remarkable identity, and reason whether the plot gains, loses, or keeps its surface area compared to the original plot of $x^2\text{ m}^2$.
2. b) Write the formula for the area in Option B as the square of a binomial and expand it. If the original plot had a side length of $x = 20\text{ m}$, calculate the exact difference in square meters between the surface area of Option B and that of Option A.

Expand the following expressions using algebraic identities and collect like terms wherever possible:

1. a) $\left( \frac{3}{2}x^2 - \frac{4}{5} \right)^2$

1. b) $(2x^3 + 5y)^2 - (2x^3 - 5y)^2$

1. c) $(x^2 - 3)(x^2 + 3) - (x^2 - 1)^2$

Find the missing term in each expression so that the trinomial is the result of the square of a sum or a difference (a perfect square). Then, write the corresponding power:

1. a) $16x^4 + \dots + 9y^2$

1. b) $\frac{1}{4}x^2 - 5x + \dots$

1. c) $49a^6 - 28a^3b + \dots$

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

An app designer is creating a square icon for a social network. The original design has a side length of $x$ pixels. The client asks for two changes: first, increase the side length by $4$ pixels to add a decorative border; second, create a reduced version where each side measures $4$ pixels less than the original.

Find the polynomial that represents the difference in area between the enlarged icon and the reduced icon in terms of $x$.

Prove algebraically that the difference between the squares of two consecutive integers is always equal to the sum of these numbers.

(Hint: call the first number $x$ and the next one $x+1$).

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 $2x-3$.

Evaluate the numerical value of the expression $E = (3x + 4y)^2 - (3x - 4y)^2$ for the values $x = \frac{1}{3}$ and $y = 5$.

Tip: Simplify the algebraic expression using algebraic identities before substituting the values.

A solar panel manufacturer has discovered that the efficiency of its panels improves if a cell configuration is used that follows the difference of two squares identity (the product of a sum and a difference).

If the total area of a panel is given by the polynomial $A(x) = 16x^4 - 81y^2$, what are the algebraic expressions that represent the length and width of the panel?

In a ceramic tile factory in Onda (Castellón), they design square tiles whose area is given by the trinomial $9x^2 + kx + 16$, where $k$ is an integer.

1. a) Deduce the two possible values that the coefficient $k$ can take so that the expression represents exactly the area of a square tile with side $(ax + b)$ or $(ax - b)$, with $a$ and $b$ being positive integers. In each case, write the associated squared binomial.
2. b) Taking the positive value of $k$ found, the designers are asked to cut out an interior decorative area by subtracting $25$ units from the total area, obtaining the expression $(3x + 4)^2 - 25$. Without expanding the squares, factor this last expression as the product of two first-degree binomials by applying the sum and difference identity, and simplify the numerical terms of each factor.

In a debate in Mathematics class, Pablo states: "The square of the sum of two numbers, $(a + b)^2$, always gives a result greater than the sum of their squares, $a^2 + b^2$, because when expanding it we always add the term $2 \cdot a \cdot b$".

1. a) Expand $(a + b)^2$ and calculate the algebraic difference $(a + b)^2 - (a^2 + b^2)$ in simplified form. What algebraic term completely determines whether the result is greater, less, or equal?
2. b) Prove that Pablo's statement is false by providing a numerical counterexample with integers where $(a + b)^2 < a^2 + b^2$ holds. Next, investigate whether it is possible to find two non-zero numbers such that $(a + b)^2 = a^2 + b^2$, and justify your conclusion.
3. c) State precisely what mathematical condition the signs of $a$ and $b$ must satisfy for $(a + b)^2 > a^2 + b^2$, when it happens that $(a + b)^2 = a^2 + b^2$, and when it happens that $(a + b)^2 < a^2 + b^2$.

Unit challenge. Design the sustainable packaging of the future! A logistics company wants to redesign its cuboidal boxes (polyhedra) to minimise the amount of cardboard used.

1. Real measurement: Use a ruler to measure the actual dimensions (length, width, and height) of a package you have at home and calculate its approximate volume.

2. Algebraic modelling: Propose a new box with a square base of side $(x+4)\text{ cm}$ and height $2x\text{ cm}$. Express using algebraic language: the binomial of a base edge, the polynomial of the total surface area of the polyhedron by expanding the special product $(x+4)^2$, and the volume polynomial. Perform the operations and simplify the polynomials.

3. Digital simulation: In a spreadsheet, generate a table evaluating the surface area and volume formulas for values of $x$ from $4$ to $15$, in increments of $0,5$. Identify the value of $x$ whose volume is closest to that of your actual package.

4. Reasoned proposal: Write a brief report including the table, the algebraic derivation, and a conclusion justifying which box makes the best use of material.

The following will be assessed:

• Correct translation into algebraic language and expansion of special products.

• Precision in operations with monomials and polynomials associated with the polyhedron.

• Rigorous use of the spreadsheet to tabulate and compare with real data.

• Clarity and mathematical justification of the final proposal.