Find the general expansion of using geometry, as we did for .

Understand the Question Geometrically, represents the area of a square with side length . We start with a large square of side length (total area ). To find the area of the square, we subtract two rectangular strips of dimensions . Since the corner square of side (area ) is removed twice, we must add back once to get the exact area. Step 1 · Area of the Large Square Consider a large square of side length . We aim to determine the area of the smaller square of side length , which is . Step 2 · Remove the First Rectangle Remove a vertical rectangular strip of length and width from the right side of the large square. Step 3 · Remove the Second Rectangle Now remove a horizontal rectangular strip of length and width along the bottom edge of the original large square. Subtracting both strips gives: Step 4 · Add Back the Overlapping Corner Square The small square at the bottom-right corner of side length was subtracted twice (once with each strip). Adding back once gives the exact area of the square of side : Answer > Common Mistakes Forgetting the term: Writing or , forgetting that the corner square of area is subtracted twice and must be added back once. Sign of the middle term: Mistakenly writing instead of .

Distributivity and Algebra | IT

Question 24

Find the general expansion of $(a - b)^2$ using geometry, as we did for $55^2$ .

Solution

Understand the Question

Geometrically, $(a - b)^2$ represents the area of a square with side length $(a - b)$ .
We start with a large square of side length $a$ (total area $= a^2$ ).
To find the area of the $(a - b) \times (a - b)$ square, we subtract two rectangular strips of dimensions $a \times b$ .
Since the corner square of side $b$ (area $= b^2$ ) is removed twice, we must add $b^2$ back once to get the exact area.

Step 1 · Area of the Large Square

Consider a large square of side length $a$ . Diagram 1

$\text{Area of large square} = a \times a = a^2$

We aim to determine the area of the smaller square of side length $(a - b)$ , which is $(a - b)^2$ .

Step 2 · Remove the First Rectangle

Remove a vertical rectangular strip of length $a$ and width $b$ from the right side of the large square. Diagram 2

$\text{Area removed (first time)} = a \times b = ab$

$\text{Remaining area} = a^2 - ab$

Step 3 · Remove the Second Rectangle

Now remove a horizontal rectangular strip of length $a$ and width $b$ along the bottom edge of the original large square. Diagram 3

$\text{Area removed (second time)} = a \times b = ab$

Subtracting both strips gives: $a^2 - ab - ab$

Step 4 · Add Back the Overlapping Corner Square

The small square at the bottom-right corner of side length $b$ was subtracted twice (once with each strip). Diagram 4

$\text{Area of small corner square} = b \times b = b^2$

Adding back $b^2$ once gives the exact area of the square of side $(a - b)$ :

\begin{aligned} (a - b)^2 &= a^2 - ab - ab + b^2 \\ &= a^2 - 2ab + b^2 \end{aligned}

Answer

$(a - b)^2 = a^2 - 2ab + b^2$

Common Mistakes

Forgetting the $+b^2$ term: Writing $(a - b)^2 = a^2 - 2ab$ or $(a - b)^2 = a^2 - b^2$ , forgetting that the corner square of area $b^2$ is subtracted twice and must be added back once.
Sign of the middle term: Mistakenly writing $+2ab$ instead of $-2ab$ .

More questions in IT

Context: Consider the multiplication of two numbers, say, $23 \times 27$ .

Q. By how much does the product increase if the first number (23) is increased by 1?

Context: Consider the multiplication of two numbers, say, $23 \times 27$ .

Q. What if the second number (27) is increased by 1?

Context: Consider the multiplication of two numbers, say, $23 \times 27$ .

Q. How about when both numbers are increased by 1?

Context: Consider the multiplication of two numbers, say, $23 \times 27$ .

Q. Do you see a pattern that could help generalise our observations to the product of any two numbers?

What would we get if we had expanded $(a + 1)(b + 1)$ by first taking $(b + 1)$ as a single term? Try it!

Will the product always increase? Find 3 examples where the product decreases.

What happens when $a$ and $b$ are negative integers?

Check by substituting different values for $a$ and $b$ in each of the above cases. For example, $a = -5$ , $b = 8$ ; $a = -4$ , $b = -5$ ; etc.

Use Identity 1 to find how the product changes when

(i) one number is decreased by 2 and the other increased by 3;

(ii) both numbers are decreased, one by 3 and the other by 4.

Verify the answers by finding the products without converting the subtractions to additions.

Q10

Expand (i) $(a - u)(b + v)$ , (ii) $(a - u)(b - v)$ .

Q11

Use the following multiplications to find the product of a number with 11 in a single step.

(a) $3874 \times 11$

(b) $5678 \times 11$

Q12

Describe a general rule to multiply a number (of any number of digits) by 11 and write the product in one line.

Evaluate:

(i) $94 \times 11$

(ii) $495 \times 11$

(iii) $3279 \times 11$

(iv) $4791256 \times 11$

Q13

Can we come up with a similar rule for multiplying a number by $101$ ?

Q14

Multiply 3874 by 101.

Q15

Use this to multiply $3874 \times 101$ in one line.

Q16

What could be a general rule to multiply a number by $101$ and write the product in one line? Extend this rule for multiplication by $1001$ , $10001$ , $\dots$

Q17

Use this to find (i) $89 \times 101$ , (ii) $949 \times 101$ , (iii) $265831 \times 1001$ , (iv) $1111 \times 1001$ , (v) $9734 \times 99$ and (vi) $23478 \times 999$ .

Q18

The area of a square of sidelength 60 units is 3600 sq. units ( $60^2$ ) and that of a square of sidelength 5 units is 25 sq. units ( $5^2$ ). Can we use this to find the area of a square of sidelength 65 units?

Q19

What if we write $65^2$ as $(30 + 35)^2$ or $(52 + 13)^2$ ? Draw the figures and check the area that you get.

Q20

If $a$ and $b$ are any two integers, is $(a + b)^2$ always greater than $a^2 + b^2$ ? If not, when is it greater?

Q21

Use Identity 1A to find the values of $104^2$ , $37^2$ . (Hint: Decompose 104 and 37 into sums or differences of numbers whose squares are easy to compute.)

Q22

Use Identity 1A to write the expressions for the following.

(i) $(m + 3)^2$

(ii) $(6 + p)^2$

Q23

Expand $(3j + 2k)^2$ using both the identity and by applying the distributive property.

Q24

Find the general expansion of $(a - b)^2$ using geometry, as we did for $55^2$ .

Q25

Use the identity $(a - b)^2$ to find the values of (a) $99^2$ and (b) $58^2$ .

Q26

Expand the following using both Identity 1B and by applying the distributive property

(i) $(b - 6)^2$

(ii) $(-2a + 3)^2$

(iii) $\left(7y - \dfrac{3}{4}z\right)^2$

Q27

Take a pair of natural numbers. Calculate the sum of their squares. Can you write twice this sum as a sum of two squares?

Try this with other pairs of numbers. Have you figured out a pattern?

Notice that $2(5^2 + 6^2) = (6 + 5)^2 + (6 - 5)^2$ .

Q28

Do the identities below help in explaining the observed pattern?

$(a + b)^2 = a^2 + 2ab + b^2$

$(a - b)^2 = a^2 - 2ab + b^2$

$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)$

Adding the like terms $a^2 + a^2 = 2a^2$ , $b^2 + b^2 = 2b^2$ and $2ab - 2ab = 0$ , we get

$2(a^2 + b^2) = (a + b)^2 + (a - b)^2$

Q29

Pattern 2

\begin{aligned} 9 \times 9 - 1 \times 1 &= 10 \times 8 \\ 8 \times 8 - 6 \times 6 &= 14 \times 2 \\ 7 \times 7 - 2 \times 2 &= 9 \times 5 \\ 10 \times 10 - 4 \times 4 &= 14 \times 6 \end{aligned}

Here is a related pattern. Try to describe the pattern using algebra to determine if the pattern always holds.

Q30

Use Identity 1C to calculate $98 \times 102$ , and $45 \times 55$ .

Q31

Show that $(a + b) \times (a - b) = a^2 - b^2$ geometrically.

Q32

Context: Sridharacharya (750 CE) gave an interesting method to quickly compute the squares of numbers using Identity 1C! Consider the following modified form of this identity — $a^2 = (a + b)(a - b) + b^2$

Q. Why is this identity true?

Q33

6.3 Mind the Mistake, Mend the Mistake

We have expanded and simplified some algebraic expressions below to their simplest forms.

(i) Check each of the simplifications and see if there is a mistake. (ii) If there is a mistake, try to explain what could have gone wrong. (iii) Then write the correct expression.

Q34

6.4 This Way or That Way, All Ways Lead to the Bay

Observe the pattern in the figure below. Draw the next figure in the sequence. How many circles does it have? How many total circles are there in Step 10? Write an expression for the number of circles in Step $k$ .

Q35

Context: The expression $k^2 + 2k$ gives the number of circles at Step $k$ of a pattern.

Q. Use this formula to find the number of circles in Step 15.

Q36

Consider the pattern made of square tiles in the picture below.

(i) How many square tiles are there in each figure? (ii) How many are there in Step 4 of the sequence? What about Step 10? (iii) Write an algebraic expression for the number of tiles in Step $n$ . Share your methods with the class. Can you find more than one method to arrive at the answer?

Q37

Context: Consider the pattern made of square tiles in the picture below.

Q. How many square tiles are there in each figure?

Q38

Context: Consider the pattern made of square tiles in the picture below.

Q. How many are there in Step 4 of the sequence? What about Step 10?

Q39

Context: Consider the pattern made of square tiles in the picture below.

Q. Write an algebraic expression for the number of tiles in Step $n$ . Share your methods with the class. Can you find more than one method to arrive at the answer?

Q40

By expanding both expressions, check that $(m + n)^2 - 4mn = (n - m)^2$ .

Q41

By expanding the expressions, verify that all three expressions are equivalent. If $x = 8$ and $y = 3$ , find the area of the shaded region.

Q42

Write an expression for the area of the dashed region in the figure below. Use more than one method to arrive at the answer. Substitute $p = 6$ , $r = 3.5$ , and $s = 9$ , and calculate the area.