Notable identities
If you edit a video and cut out a square of $b$ pixels in one corner of a square frame of side $a$, the area that is left is not $a^2 - b^2$ by chance: it is a notable identity working for you.
The three identities
You have already seen them in 3rd ESO. It is worth having them memorised because they come up all the time when simplifying and factorising.
Move $a$ and $b$ and see how the area of the large square is shared among the three pieces:
Expand the square of the following sum using the corresponding algebraic identity: $(4x + 7)^2$.
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: $(5a^2 - 2b)^2$.
Expand the square of the following difference applying the appropriate algebraic identity: $(5a^2 - 2b)^2$.
Square of a difference and sum times difference
The square of a difference looks just like the square of a sum, but you have to take back the little square that is subtracted twice. The sum times difference, on the other hand, always gives a difference of squares, with no middle term.
Calculate the product of the sum and difference for the following expression: $(3m^3 + 8n)(3m^3 - 8n)$.
Calculate the product of the sum and difference for the following expression: $(3m^3 + 8n)(3m^3 - 8n)$.
Apply algebraic identities to mentally calculate the exact value of $53^2$ using the expression $(50 + 3)^2$.
Apply algebraic identities to mentally calculate the exact value of $53^2$ using the expression $(50 + 3)^2$.
Using them in reverse: factorising
If you recognise the "shape" of an identity in a polynomial that has already been expanded, you can go back along the same path and write it as a product. It is the same idea you will use later with the common factor, but here the pattern is always one of these three:
Factorise the following perfect square trinomial, expressing it as the square of a binomial: $x^2 + 14x + 49$.
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: $25y^2 - 60y + 36$.
Express the following polynomial as the square of a difference: $25y^2 - 60y + 36$.
Factorise the following difference of two squares: $81z^2 - 100w^6$.
Factorise the following difference of two squares: $81z^2 - 100w^6$.
Applications: mental arithmetic and geometry
Identities are not just an algebra exercise: they help you calculate faster and model areas where one part varies.
In geometry, for example, the walkable area of a square plaza of side $3x+5$ with a square fountain of side $x-2$ in one corner is a difference of squares:
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.…
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.
Fully simplify the following combined expression by collecting like terms: $(2x + 5)^2 - (2x - 5)^2$.
Fully simplify the following combined expression by collecting like terms: $(2x + 5)^2 - (2x - 5)^2$.
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.
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.
Back to the start: When you cut a square corner of side $b$ from a frame of side $a$, the remaining area is $a^2 - b^2$, which, thanks to the sum times difference, can be broken down into a rectangle with sides $(a + b)$ and $(a - b)$.