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.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Recent Articles

1. Successor and Predecessor | Successor of a Whole Number | Predecessor

May 24, 24 06:42 PM

The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

2. Counting Natural Numbers | Definition of Natural Numbers | Counting

May 24, 24 06:23 PM

Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

3. Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

May 24, 24 06:22 PM

The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

4. Math Questions Answers | Solved Math Questions and Answers | Free Math

May 24, 24 05:37 PM

In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students