MathematicsClass 8CBSE

The Middle Term Everyone Forgets When Squaring a Bracket

Learn the identities for a sum squared and a difference squared, use the difference of two squares to compute products instantly, and learn to spot and correct a wrong expansion.

Why is a plus b, all squared, not just a squared plus b squared?

Because squaring a bracket means multiplying it by itself, which produces a middle term as well. Test it with numbers:

(3 + 4)^{2} = 7^{2} = 49

, while

3^{2} + 4^{2} = 9 + 16 = 25

.

The missing

2 ab = 2 \times 3 \times 4 = 24

makes up the difference, since

9 + 24 + 16 = 49

. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: the three standard identities and how to spot a wrong expansion.

Formula

How do you expand a sum squared?

The identity is:

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

It comes straight from multiplying the bracket by itself:

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

The two middle products are identical, which is where the 2 comes from.

Worked examples.

(x + 5)^{2} = x^{2} + 2 (x) (5) + 25 = x^{2} + 10 x + 25

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

(3 a + 4 b)^{2} = 9 a^{2} + 24 ab + 16 b^{2}

Used on numbers, it makes awkward squares easy:

10 2^{2} = (100 + 2)^{2} = 10000 + 400 + 4 = 10404

5 1^{2} = (50 + 1)^{2} = 2500 + 100 + 1 = 2601

2 5^{2} = (20 + 5)^{2} = 400 + 200 + 25 = 625

The term to watch is the middle one, and the trap is the coefficient. In

(2 x + 3)^{2}

the middle term is

2 \times 2 x \times 3 = 12 x

, not

6 x

— the

2

from the identity multiplies the whole of

2 x

and

3

. Writing

4 x^{2} + 6 x + 9

is the standard error.

How do you expand a difference squared?

The identity is the same shape with one sign changed:

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

Only the middle term becomes negative. The last term stays positive, because

(- b) \times (- b) = + b^{2}

.

Worked examples.

(x - 4)^{2} = x^{2} - 2 (x) (4) + 16 = x^{2} - 8 x + 16

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

(5 a - 2 b)^{2} = 25 a^{2} - 20 ab + 4 b^{2}

Used on numbers:

9 8^{2} = (100 - 2)^{2} = 10000 - 400 + 4 = 9604

4 9^{2} = (50 - 1)^{2} = 2500 - 100 + 1 = 2401

9 7^{2} = (100 - 3)^{2} = 10000 - 600 + 9 = 9409

The sign of the last term is what students get wrong. In

(x - 4)^{2}

the final term is

+ 16

, not

- 16

, because a negative number squared is positive. So the expansion has one minus sign and two plus signs — check that pattern on every answer.

The two identities can be remembered as one:

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

, where only the middle sign follows the bracket.

How does the difference of two squares speed up multiplication?

The identity is:

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

Expanding shows why the middle terms disappear:

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

The

- ab

and

+ ab

cancel exactly.

Worked examples in algebra.

(x + 7) (x - 7) = x^{2} - 49

(2 x + 5) (2 x - 5) = 4 x^{2} - 25

(3 a + 4 b) (3 a - 4 b) = 9 a^{2} - 16 b^{2}

Used on numbers, it is the fastest of the three. Look for two numbers equally spaced either side of a round number:

103 \times 97 = (100 + 3) (100 - 3) = 10000 - 9 = 9991

52 \times 48 = (50 + 2) (50 - 2) = 2500 - 4 = 2496

45 \times 35 = (40 + 5) (40 - 5) = 1600 - 25 = 1575

106 \times 94 = (100 + 6) (100 - 6) = 10000 - 36 = 9964

It also factorises any difference of two squares:

x^{2} - 81 = (x + 9) (x - 9), 49 - y^{2} = (7 + y) (7 - y)

The condition for the numerical shortcut is that the two numbers must be the same distance from the midpoint. So

103 \times 97

works because both are 3 away from 100, but

103 \times 96

does not — checking that the offsets match is the step to take before reaching for this identity.

How do you spot and correct a wrong expansion?

Substitute numbers into both the original expression and the answer. If they disagree, the expansion is wrong — and the difference tells you what is missing.

Error 1. A student writes

(x + 5)^{2} = x^{2} + 25

.

Test with

x = 1

: the original is

(1 + 5)^{2} = 36

, while the answer gives

1 + 25 = 26

. They differ by 10.

The middle term

2 ab = 2 (x) (5) = 10 x

was omitted. The correct expansion is

x^{2} + 10 x + 25

and at

x = 1

that gives

1 + 10 + 25 = 36

. Correct.

Error 2. A student writes

(x - 3)^{2} = x^{2} - 6 x - 9

.

Test with

x = 0

: the original is

(- 3)^{2} = 9

, while the answer gives

- 9

. The last term's sign is wrong, since

(- 3)^{2} = + 9

. The correct expansion is

x^{2} - 6 x + 9

.

Error 3. A student writes

(2 x + 3)^{2} = 4 x^{2} + 6 x + 9

.

Test with

x = 1

: the original is

5^{2} = 25

, while the answer gives

4 + 6 + 9 = 19

. The middle term should be

2 (2 x) (3) = 12 x

, giving

4 x^{2} + 12 x + 9

, which at

x = 1

25

. Correct.

An alternative method always exists, and showing it confirms the result. Instead of the identity, expand

(2 x + 3)^{2}

as a plain product of two brackets:

(2 x + 3) (2 x + 3) = 4 x^{2} + 6 x + 6 x + 9 = 4 x^{2} + 12 x + 9

The same answer by a different route.

Substitution is the tool worth carrying away. Any single value of

x

that makes the two sides disagree proves the expansion wrong, and

x = 1

x = 0

is usually the quickest to test — which is why it is worth doing before writing the answer down.

Exam tip

Exam tip: checking your expansion with a number

Identity questions are quick marks, and one substitution protects all of them.

After expanding, **put

x = 1

into the original and into your answer. If they match, the expansion is almost certainly right; if not, you have caught the error before it cost anything.

Write the identity before** substituting: *

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

*. That line is marked on its own.

For the middle term, multiply the whole of each part — in

(2 x + 3)^{2}

it is

2 \times 2 x \times 3 = 12 x

, not

6 x

.

Check the sign pattern: a difference squared has one minus sign, in the middle, and a positive last term.

And before using

a^{2} - b^{2}

on numbers, confirm the two are the same distance from the round midpoint.

Did you know

Why does squaring a bracket produce three terms rather than two?

Because two terms multiplied by two terms give four products, and two of them turn out to be the same.

Expanding

(a + b) (a + b)

produces

a^{2}

, then

ab

, then

ba

, then

b^{2}

. The two middle products are identical, so they combine into a single

2 ab

— leaving three terms rather than four.

Picture it as area: a square of side

a + b

splits into a square of

a^{2}

, a square of

b^{2}

, and two rectangles each measuring

a

b

. Those two rectangles are precisely the

2 ab

that the careless expansion throws away.

Key takeaways

Algebraic identities: quick revision

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

— the middle term exists because two identical products combine, so

(3 + 4)^{2} = 49

, not

25

.
-

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

— only the middle sign changes, and the last term stays positive.
-

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

, because the middle products cancel.
- On numbers:

10 2^{2} = 10404

9 8^{2} = 9604

103 \times 97 = 9991

and

52 \times 48 = 2496

— but the two factors must be equally spaced from the midpoint.
- The identity also factorises:

x^{2} - 81 = (x + 9) (x - 9)

.
- **Check any expansion by substituting

x = 1

** — a mismatch proves it wrong, and expanding as a plain product of two brackets is the alternative method that confirms the right answer.

You will remember all of this far better after answering five questions on it than after reading it twice.

Why is a plus b, all squared, not just a squared plus b squared?

How do you expand a sum squared?

How do you expand a difference squared?

How does the difference of two squares speed up multiplication?

How do you spot and correct a wrong expansion?

Exam tip: checking your expansion with a number

Why does squaring a bracket produce three terms rather than two?

Algebraic identities: quick revision

Related study notes

Ready to put this into practice?