Notable identities
What is $52 \cdot 48$ without a calculator? If you notice that $52 = 50 + 2$ and $48 = 50 - 2$, the calculation becomes $50^2 - 2^2 = 2\,500 - 4 = 2\,496$. Notable identities are shortcuts like this: they come up so often that it is worth knowing them by heart.
The square of a sum
Change $a$ and $b$ and see how the area of the big square is split up:
Lucía wants to design a square rug for her bedroom. The side of the rug measures $(x + 4)$ metres. a) Write the expression for the area of the rug as the square of a…
Lucía wants to design a square rug for her bedroom. The side of the rug measures $(x + 4)$ metres.
- a) Write the expression for the area of the rug as the square of a binomial.
- b) Expand the binomial to obtain the polynomial representing the area.
The square of a difference
Fill in the blanks so that the following equations are correct: a) $(x + 4)^2 = x^2 + \dots x + 16$ b) $(a - \dots)^2 = a^2 - 6a + 9$ c) $(y + 5)(y - 5) = y^2 - \dots$
Fill in the blanks so that the following equations are correct:
- a) $(x + 4)^2 = x^2 + \dots x + 16$
- b) $(a - \dots)^2 = a^2 - 6a + 9$
- c) $(y + 5)(y - 5) = y^2 - \dots$
The difference of two squares
Match each algebraic identity product in your exercise book with its corresponding expanded expression: A. $(x + 2)^2$ B. $(x - 2)^2$ C. $(x + 2)(x - 2)$ 1. $x^2 - 4$ 2.…
Match each algebraic identity product in your exercise book with its corresponding expanded expression:
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A. $(x + 2)^2$
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B. $(x - 2)^2$
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C. $(x + 2)(x - 2)$
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$x^2 - 4$
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$x^2 + 4x + 4$
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$x^2 - 4x + 4$
Spotting and correcting mistakes
The three identities are easily mixed up. Before applying one, look at whether there is a plus sign, a minus sign, or both (one in each factor).
| Expression | Identity | Result |
|---|---|---|
| $(x+4)^2$ | square of a sum | $x^2 + 8x + 16$ |
| $(x-5)(x+5)$ | difference of two squares | $x^2 - 25$ |
| $(2a-3)^2$ | square of a difference | $4a^2 - 12a + 9$ |
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…
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:
- a) $(x + 4)^2 = x^2 + 16$
- b) $(x - 5) \cdot (x + 5) = x^2 - 25$
- c) $(2a - 3)^2 = 4a^2 - 12a + 9$
In reverse: factorising with identities
Recognising an identity also helps you go from the expanded polynomial back to the product, that is, to factorise.
$$x^2 - 9 = (x+3)(x-3) \qquad x^2 + 6x + 9 = (x+3)^2$$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…
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.
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…
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:
- a) $52 \cdot 48$ (Hint: consider $50 + 2$ and $50 - 2$)
- b) $99 \cdot 101$
- c) $35 \cdot 25$