Algebraic identities

Q: What are the standard algebraic identities?

(a+b)^2, (a-b)^2, a^2-b^2 and (x+a)(x+b), as listed above.

Q: How do identities help with arithmetic?

They turn awkward products into easy ones, such as 99^2 = (100-1)^2 = 9801 and 51 \times 49 = 50^2 - 1 = 2499.

An algebraic identity is an equation that is true for every value of the letters. The standard identities let you expand brackets, factorise expressions and do some arithmetic quickly in your head.

Formulas

Square of a sum

(a + b)^{2} = a^{2} + 2 ab + b^{2}

Square of a difference

(a - b)^{2} = a^{2} - 2 ab + b^{2}

Difference of two squares

a^{2} - b^{2} = (a + b) (a - b)

Product of two binomials

(x + a) (x + b) = x^{2} + (a + b) x + ab

Worked examples

Example 1. Expand

(3 x + 2)^{2}

Use $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ with $a = 3 x, b = 2$ .
$= 9 x^{2} + 2 (3 x) (2) + 4 = 9 x^{2} + 12 x + 4$ .

Answer:

9 x^{2} + 12 x + 4

Example 2. Find

10 3^{2}

using an identity.

$103 = 100 + 3, so 10 3^{2} = 10 0^{2} + 2 (100) (3) + 3^{2}$ .
$= 10000 + 600 + 9 = 10609$ .

Answer: 10 609

Example 3. Find

98 \times 102

using an identity.

$98 \times 102 = (100 - 2) (100 + 2) = 10 0^{2} - 2^{2}$ .
$= 10000 - 4 = 9996$ .

Answer: 9996

Example 4. Factorise

25 x^{2} - 16

$25 x^{2} = (5 x)^{2}$ and $16 = 4^{2}$ , a difference of two squares.
$25 x^{2} - 16 = (5 x + 4) (5 x - 4)$ .

Answer:

(5 x + 4) (5 x - 4)

Common mistakes

Writing $(a + b)^{2} = a^{2} + b^{2}$ . The middle term $2 ab$ is always there.
Getting the sign of the middle term wrong. It is $+ 2 ab$ for a sum and $- 2 ab$ for a difference.
Trying to factorise a sum of two squares like $a^{2} + b^{2}$ with the difference-of-squares identity. It only works for a difference.

Practice questions with answers

1. Which expression is the correct expansion of

(a + b)^{2}

A $a^{2} + b^{2}$
B $a^{2} + 2 ab + b^{2}$
C $a^{2} - 2 ab + b^{2}$
D $a^{2} + ab + b^{2}$

Show answer

Answer: B)

a^{2} + 2 ab + b^{2}

The standard identity is

(a + b)^{2} = a^{2} + 2 ab + b^{2}

2. Expand

(x + 3)^{2}

A $x^{2} + 6 x + 6$
B $x^{2} + 6 x + 9$
C $x^{2} + 9$
D $x^{2} + 3 x + 9$

Show answer

Answer: B)

x^{2} + 6 x + 9

(a + b)^{2} = a^{2} + 2 ab + b^{2} : x^{2} + 2 (x) (3) + 9 = x^{2} + 6 x + 9

. Leaving out the middle term

6 x

is the most common slip.

3. Multiply:

4 x \times (2 x + 3)

A $8 x^{2} + 12 x$
B $8 x + 12 x$
C $6 x^{2} + 7 x$
D $8 x^{2} + 12$

Show answer

Answer: A)

8 x^{2} + 12 x

Distribute:

4 x \times 2 x + 4 x \times 3 = 8 x^{2} + 12 x

4. Priya checks that for any number

x

, the expression

x^{2} - (x - 1) (x + 1)

is always the same value.
What is that constant value?

Show answer

Answer: C) 1

(x - 1) (x + 1) = x^{2} - 1

, so

x^{2} - (x^{2} - 1) = 1

for every

x

. The constant value is 1.

5. Factorise

27 x^{3} + y^{3} + z^{3} - 9 x yz

A $(9 x + y + z) (x^{2} + y^{2} + z^{2} - x y - yz - x z)$
B $(3 x + y + z)^{3}$
C $(3 x + y + z) (9 x^{2} + y^{2} + z^{2} - 3 x y - yz - 3 x z)$
D $(3 x + y + z) (9 x^{2} + y^{2} + z^{2} + 3 x y + yz + 3 x z)$

Show answer

Answer: C)

(3 x + y + z) (9 x^{2} + y^{2} + z^{2} - 3 x y - yz - 3 x z)

With

a = 3 x, b = y, c = z : 3 ab c = 9 x yz

, so use

a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)

6. Neha writes

(2 x + 3)^{2} = 4 x^{2} + 9

. Substituting

x = 1

into both sides, what does the check show?

ABoth sides give 13, so Neha is correct
BLeft side is 25 but right side is 13 — the middle term $12 x$ is missing
CLeft side is 13 but right side is 25 — the 3 should not be squared
DLeft side is 25 and right side is 25, so the check proves nothing

Show answer

Answer: B) Left side is 25 but right side is 13 — the middle term

12 x

is missing

x = 1

the left side is

(2 + 3)^{2} = 25

, while Neha's right side gives 4 + 9 = 13. The sides disagree, so her expansion is wrong. The correct one is

(2 x + 3)^{2} = 4 x^{2} + 2 (2 x) (3) + 9 = 4 x^{2} + 12 x + 9

, which at

x = 1

gives 4 + 12 + 9 = 25 and matches.

(a + b)^{2}

always carries a middle term.

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Frequently asked questions

What is an algebraic identity?

An equation that holds for all values of the variables, unlike an ordinary equation that holds only for particular values.

What are the standard algebraic identities?

$(a + b)^{2}, (a - b)^{2}, a^{2} - b^{2}$ and $(x + a) (x + b)$ , as listed above.

How do identities help with arithmetic?

They turn awkward products into easy ones, such as $9 9^{2} = (100 - 1)^{2} = 9801$ and $51 \times 49 = 5 0^{2} - 1 = 2499$ .

How do I know which identity to use?

Look at the shape. A bracket squared means the first two; two squares being subtracted means $a^{2} - b^{2}$ ; two brackets with the same variable means $(x + a) (x + b)$ .

Algebraic identities

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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