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

Distributivity and Algebra | IT

Question 24

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

Solution

We will use the areas of squares and rectangles to understand how $(a-b)^2$ expands.

Step 1 — Start with a large square

Let us imagine a big square. Its side length is 'a'. The total area of this large square is $a \times a$ .

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

We want to find the area of a smaller square. This smaller square will have a side length of $(a-b)$ .

Diagram 1

Step 2 — Remove the first rectangle

From our big square, let us remove a rectangle. This rectangle has a length of 'a' and a width of 'b'. We can imagine removing it from the right side of the large square. The area of this removed rectangle is $a \times b$ .

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

The area remaining after removing this first rectangle is $a^2 - ab$ .

Diagram 2

Step 3 — Remove the second rectangle

Now, let us remove another rectangle. This rectangle also has a length of 'a' and a width of 'b'. We can imagine removing it from the bottom side of the original large square. The area of this second removed rectangle is also $a \times b$ .

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

If we simply subtract both these areas, we get $a^2 - ab - ab$ . However, we have made a mistake in this simple subtraction.

Diagram 3

Step 4 — Correct for the double subtraction

When we removed the first rectangle ( $a \times b$ ) and the second rectangle ( $a \times b$ ), we subtracted a small corner square twice. This small square is located at the bottom-right corner. Its side length is 'b' (the width of the first strip and the height of the second strip). The area of this small square is $b \times b$ .

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

Since we subtracted this $b^2$ area twice, we need to add it back once. This will give us the correct area of the square with side $(a-b)$ .

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

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

This is the general expansion of $(a-b)^2$ using geometry.

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

Diagram 4

More questions in IT

Context: Consider the multiplication of two numbers, say, 23 × 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 × 27.

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

Context: Consider the multiplication of two numbers, say, 23 × 27.

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

Context: Consider the multiplication of two numbers, say, 23 × 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, ...

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

Q27

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

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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

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(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

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Q36

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Q37

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Q38

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Q39

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Q40

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

Q41

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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.