# Expansion of (a ± b)$$^{2}$$

A binomial is an algebraic expression which has exactly two terms, for example, a ± b. Its power is indicated by a superscript. For example, (a ± b)2 is a power of the binomial a ± b, the index being 2.

A trinomial is an algebraic expression which has exactly three terms, for example, a ± b ± c. Its power is also indicated by a superscript. For example, (a ± b ± c)3 is a power of the trinomial a ± b ± c, whose index is 3.

Expansion of (a ± b)2

(a +b)$$^{2}$$

= (a + b)(a + b)

= a(a + b) + b(a+ b)

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

= a$$^{2}$$ + 2ab + b$$^{2}$$.

(a - b)$$^{2}$$

= (a - b)(a - b)

= a(a - b) - b(a - b)

= a$$^{2}$$ - ab - ab + b$$^{2}$$

= a$$^{2}$$ - 2ab + b$$^{2}$$.

Therefore, (a + b)$$^{2}$$ + (a - b)$$^{2}$$

= a$$^{2}$$ + 2ab + b$$^{2}$$ + a$$^{2}$$ - 2ab + b$$^{2}$$

= 2(a$$^{2}$$ + b$$^{2}$$), and

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

= a$$^{2}$$ + 2ab + b$$^{2}$$ - {a$$^{2}$$ - 2ab + b$$^{2}$$}

= a$$^{2}$$ + 2ab + b$$^{2}$$ - a$$^{2}$$ + 2ab - b$$^{2}$$

= 4ab.

Corollaries:

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

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

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

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

(v) (a - b)$$^{2}$$ = (a + b)$$^{2}$$ - 4ab

(vi) (a + b)$$^{2}$$ = (a - b)$$^{2}$$ + 4ab

(vii) (a + $$\frac{1}{a}$$)$$^{2}$$ = a$$^{2}$$ + 2a ∙ $$\frac{1}{a}$$ + ($$\frac{1}{a}$$)$$^{2}$$ = a$$^{2}$$ + $$\frac{1}{a^{2}}$$ + 2

(viii) (a - $$\frac{1}{a}$$)$$^{2}$$ = a$$^{2}$$ - 2a ∙ $$\frac{1}{a}$$ + ($$\frac{1}{a}$$)$$^{2}$$ = a$$^{2}$$ + $$\frac{1}{a^{2}}$$ - 2

Thus, we have

1. (a +b)$$^{2}$$ = a$$^{2}$$ + 2ab + b$$^{2}$$.

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

3. (a + b)$$^{2}$$ + (a - b)$$^{2}$$  = 2(a$$^{2}$$ + b$$^{2}$$)

4. (a + b)$$^{2}$$ - (a - b)$$^{2}$$ = 4ab.

5. (a + $$\frac{1}{a}$$)$$^{2}$$ = a$$^{2}$$ + $$\frac{1}{a^{2}}$$ + 2

6. (a - $$\frac{1}{a}$$)$$^{2}$$ = a$$^{2}$$ + $$\frac{1}{a^{2}}$$ - 2

Solved Example on Expansion of (a ± b)2

1. Expand (2a + 5b)$$^{2}$$.

Solution:

(2a + 5b)$$^{2}$$

= (2a)$$^{2}$$ + 2 ∙ 2a ∙ 5b + (5b)$$^{2}$$

= 4a$$^{2}$$ + 20ab + 25b$$^{2}$$

2. Expand (3m - n)$$^{2}$$

Solution:

(3m - n)$$^{2}$$

= (3m)$$^{2}$$ - 2 ∙ 3m ∙ n + n$$^{2}$$

= 9m$$^{2}$$ - 6mn + n$$^{2}$$

3. Expand (2p + $$\frac{1}{2p}$$)$$^{2}$$

Solution:

(2p + $$\frac{1}{2p}$$)$$^{2}$$

= (2p)$$^{2}$$ + 2 ∙ 2p ∙ $$\frac{1}{2p}$$ + ($$\frac{1}{2p}$$)$$^{2}$$

= 4p$$^{2}$$ + 2 + $$\frac{1}{4p^{2}}$$

4. Expand (a - $$\frac{1}{3a}$$)$$^{2}$$

Solution:

(a - $$\frac{1}{3a}$$)$$^{2}$$

= a$$^{2}$$ - 2 ∙ a ∙ $$\frac{1}{3a}$$ + ($$\frac{1}{3a}$$)$$^{2}$$

= a$$^{2}$$ - $$\frac{2}{3}$$ + $$\frac{1}{9a^{2}}$$.

5. If a + $$\frac{1}{a}$$ = 3, find (i) a$$^{2}$$ + $$\frac{1}{a^{2}}$$ and (ii) a$$^{4}$$ + $$\frac{1}{a^{4}}$$

Solution:

We know, x$$^{2}$$ + y$$^{2}$$ = (x + y)$$^{2}$$ – 2xy.

Therefore, a$$^{2}$$ + $$\frac{1}{a^{2}}$$

= (a + $$\frac{1}{a}$$)$$^{2}$$ – 2 ∙ a ∙ $$\frac{1}{a}$$

= 3$$^{2}$$ – 2

= 9 – 2

= 7.

Again, Therefore, a$$^{4}$$ + $$\frac{1}{a^{4}}$$

= (a$$^{2}$$ + $$\frac{1}{a^{2}}$$)$$^{2}$$ – 2 ∙ a$$^{2}$$ ∙ $$\frac{1}{a^{2}}$$

= 7$$^{2}$$ – 2

= 49 – 2

= 47.

6. If a - $$\frac{1}{a}$$ = 2, find a$$^{2}$$ + $$\frac{1}{a^{2}}$$

Solution:

We know, x$$^{2}$$ + y$$^{2}$$ = (x - y)$$^{2}$$ + 2xy.

Therefore, a$$^{2}$$ + $$\frac{1}{a^{2}}$$

= (a - $$\frac{1}{a}$$)$$^{2}$$ + 2 ∙ a ∙ $$\frac{1}{a}$$

= 2$$^{2}$$ + 2

= 4 + 2

= 6.

7. Find ab if a + b = 6 and a – b = 4.

Solution:

We know, 4ab = (a + b)$$^{2}$$ – (a – b)$$^{2}$$

= 6$$^{2}$$ – 4$$^{2}$$

= 36 – 16

= 20

Therefore, 4ab = 20

So, ab = $$\frac{20}{4}$$ = 5.

8. Simplify: (7m + 4n)$$^{2}$$ + (7m - 4n)$$^{2}$$

Solution:

(7m + 4n)$$^{2}$$ + (7m - 4n)$$^{2}$$

= 2{(7m)$$^{2}$$ + (4n)$$^{2}$$}, [Since (a + b)$$^{2}$$ + (a – b)$$^{2}$$ = 2(a$$^{2}$$ + b$$^{2}$$)]

= 2(49m$$^{2}$$+ 16n$$^{2}$$)

= 98m$$^{2}$$ + 32n$$^{2}$$.

9. Simplify: (3u + 5v)$$^{2}$$ - (3u - 5v)$$^{2}$$

Solution:

(3u + 5v)$$^{2}$$ - (3u - 5v)$$^{2}$$

= 4(3u)(5v), [Since (a + b)$$^{2}$$ - (a – b)$$^{2}$$ = 4ab]

= 60uv.