Exercises: Algebraic identities

Expand the square of the following sum using the corresponding algebraic identity: $(4x + 7)^2$.

Expand the square of the following difference applying the appropriate algebraic identity: $(5a^2 - 2b)^2$.

Calculate the product of the sum and difference for the following expression: $(3m^3 + 8n)(3m^3 - 8n)$.

Factorise the following perfect square trinomial, expressing it as the square of a binomial: $x^2 + 14x + 49$.

Express the following polynomial as the square of a difference: $25y^2 - 60y + 36$.

Factorise the following difference of two squares: $81z^2 - 100w^6$.

Determine the missing term so that the following polynomial is a perfect square trinomial: $9x^2 - \dots + 64y^2$.

Apply algebraic identities to mentally calculate the exact value of $53^2$ using the expression $(50 + 3)^2$.

Lucía is calculating the area of a square metal sheet whose side measures $(2x - 3)\text{ meters}$ and needs to expand the algebraic expression for a workshop report. In her notebook, she writes the following:

$(2x - 3)^2 = (2x)^2 - 3^2 = 4x^2 - 9$

1. a) Identify the two conceptual mistakes Lucía made when applying the notable identity.
2. b) Explain to Lucía why the expansion of the square of a difference must result in a trinomial and not a binomial, using the distributive property.
3. c) Write the correct expansion and check that your result does not match Lucía's by substituting $x = 4$ into both the initial expression and both answers.

A cooperative oil mill in Jaén is going to roof an unloading area by installing photovoltaic panels on a rectangular roof that measures exactly $52\text{ m}$ long by $48\text{ m}$ wide.

1. a) Without using a calculator or paper, mentally estimate the area by rounding both measurements to $50\text{ m}$. Reason whether you expect the actual area of the roof to be greater than, less than, or equal to the $2500\text{ m}^2$ resulting from that approximation.
2. b) Write the dimensions as the product of a sum and a difference, $(50 + 2)\cdot(50 - 2)$, and calculate the exact area mentally by applying the corresponding notable identity.
3. c) Check your initial estimate by calculating the difference between the exact and estimated values. Why does this difference coincide exactly with $2^2\text{ m}^2$?

An architect designs a square town square with side length $3x + 5$ metres. Inside it, they place a fountain, which is also square with side length $x - 2$ metres. Obtain the ordered, simplified polynomial representing the walkable area of the town square.

Expand the following algebraic expression with fractional coefficients and collect like terms: $\left(\frac{2}{3}x^2 - \frac{5}{4}y\right)^2$.

Simplify the following algebraic fraction by first factorising the numerator and denominator using algebraic identities: $\frac{x^2 - 16}{x^2 - 8x + 16}$.

A company estimates that if it sets the price of a product at $150 + 3x$ euros, it will sell $150 - 3x$ units. Determine the polynomial that models total revenue and calculate what the revenue will be if $x = 10$.

Fully simplify the following combined expression by collecting like terms: $(2x + 5)^2 - (2x - 5)^2$.

A farmland plot is circular in shape with a radius of $R$ metres. It is decided to build an outer perimeter path of width $x$ metres. Express the area of the path in terms of $R$ and $x$ by expanding the expression using algebraic identities.

Expand the cube of the following difference using the corresponding formula: $(2x - 3)^3$.

Knowing that the sum of two numbers is $S = x+y = 12$ and their product is $P = xy = 35$, calculate the value of $x^2 + y^2$ without finding the individual values of $x$ and $y$. (Hint: use the expansion of $(x+y)^2$).

The urban planning department of a municipality is designing a rectangular children's playground whose area is given by the polynomial $A(x) = 25x^2 - 36$, with dimensions expressed in meters and $x > \frac{6}{5}$.

1. a) Determine the algebraic expressions for the base and the height of this playground by factoring the polynomial using the appropriate notable identity.
2. b) In a second phase, it is decided to transform the area into a completely square plaza whose area is represented by the trinomial $25x^2 + kx + 36$. Find the positive value of the parameter $k$ for this trinomial to be a perfect square and deduce the expression for the side of the plaza in terms of $x$.
3. c) If for a specific construction phase the area of the original rectangular area is $64\text{ m}^2$, calculate the numerical value of $x$, the dimensions of its sides, and the area that the new square plaza will have.

As part of the community urban garden plan in Sevilla, they start with an experimental square plot of land of side $x\text{ meters}$. Environmental technicians propose two independent modifications to this design:

• Garden A: Increase one side by $4\text{ m}$ and decrease the perpendicular side by $4\text{ m}$, obtaining a rectangular enclosure of dimensions $(x + 4)$ and $(x - 4)$ meters.

• Garden B: Increase both sides by $4\text{ m}$, obtaining a larger square plot of side $(x + 4)$ meters.

1. a) Expand and simplify the polynomials representing the area of Garden A and the area of Garden B by applying the relevant notable identities.
2. b) Obtain the ordered polynomial expressing by how many square meters the area of Garden B exceeds that of Garden A.
3. c) A neighborhood spokesperson claims that Garden A has exactly the same area as the original garden of side $x$ because "what is added to the length is taken from the width." Justify algebraically whether this statement is true or false and determine how many square meters are gained or lost compared to the initial plot.

Calculate the value or values of the parameter $k$ so that the polynomial $49x^4 + kx^2y^3 + 144y^6$ is a perfect square trinomial.

A square cardboard sheet of side length $L$ centimetres is available. To construct an open-topped box, four equal squares of side length $x$ are cut out from each corner and the flaps are folded up. Obtain the polynomial expressing the area of the base of the box and the polynomial expressing its volume.

Prove algebraically that the sum of the square of the difference of two numbers and four times their product is equal to the square of the sum of those numbers. That is, expand and simplify to prove that $(a-b)^2 + 4ab = (a+b)^2$.

Fully factorise the polynomial $16x^4 - 81y^4$, applying the difference of two squares identity as many times as possible over the set of real numbers.

Simplify the following algebraic expression, first expanding the numerators and factorising the denominators: $\frac{(a+b)^2 - (a-b)^2}{a^2b - ab^2} \cdot \frac{a-b}{4}$.

Prove that the difference between the squares of two consecutive odd numbers is always a multiple of $8$. Use the expressions $2n+1$ and $2n-1$ to represent these numbers.

Expand and fully simplify the following nested expression: $\left( (x+2)^2 - (x-2)^2 \right)^2$.

A mechanical washer is shaped like an annulus. Its outer radius is given by the expression $R = 5x + 3$ millimetres, and its inner radius by $r = 5x - 3$ millimetres. Determine the simplified polynomial that expresses the surface area of the flat face of the washer.

To calculate the numerical value of the expression $E(x) = (2x + 5)^2 - (2x - 5)^2$ when $x = 12,5$, two 4th-year ESO students propose different strategies:

• Marcos's method: Expand the square of the sum and the square of the difference independently, subtract both trinomials by combining like terms, and, finally, substitute $x = 12,5$.

• Sara's method: Consider the complete expression as a difference of squares of the form $A^2 - B^2$, transform it into the product of a sum and its difference, simplify the resulting factors, and substitute $x = 12,5$.

1. a) Apply Marcos's method, clearly indicating how signs are handled when subtracting.
2. b) Apply Sara's method by working with the structure $[(2x + 5) + (2x - 5)]\cdot[(2x + 5) - (2x - 5)]$.
3. c) Compare both procedures and justify which is more advantageous and safer for avoiding typical sign errors. Explain how this advantage would be maintained if the bases were longer polynomials, for example $(x^2 + 2x + 1)^2 - (x^2 + 2x - 1)^2$.

In a wind farm in Castilla y León, technical protocols establish a safety zone in the shape of a circular ring (the flat area contained between two concentric circles) around each wind turbine. The radius of the inner exclusion circle for machinery is $R_1 = (2x - 3)\text{ meters}$ and the radius of the outer circle marking the reach of the blades is $R_2 = (2x + 3)\text{ meters}$, with $x > 3$.

1. a) Set up the formula for the area of this circular ring by factoring out the number $\pi$: $A = \pi \cdot (R_2^2 - R_1^2)$.
2. b) Without expanding each square separately, apply the difference of squares notable identity to factor $(R_2^2 - R_1^2)$ and obtain the simplified polynomial expression for the area of the ring in terms of $x$ and $\pi$.
3. c) If the topographical survey certifies that the area of this circular ring is exactly $144\pi\text{ m}^2$, set up an equation, calculate the value of $x$, and determine the outer radius $R_2$. Reason whether a rotor diameter of $2\cdot R_2 = 30\text{ m}$ is plausible for a medium-power wind turbine in Spain.