Exploring Algebraic Identities | Exercise 4.3

Question 1

Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier.

(i) $117^2$

(ii) $78^2$

(iii) $198^2$

(iv) $214^2$

(v) $1104^2$

(vi) $1120^2$

Solution

Understand the Question

To calculate the square of a number easily without long multiplication, express the number as a sum $(a + b)$ or difference $(a - b)$ using a convenient base number (such as a multiple of $10$ , $100$ , or $1000$ ):

Sum Identity: $(a + b)^2 = a^2 + 2ab + b^2$
Difference Identity: $(a - b)^2 = a^2 - 2ab + b^2$

(i) $117^2$

Step 1 · Evaluate $117^2$ using $(a+b)^2$

Express $117$ as $(110 + 7)$ and apply $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 110$ and $b = 7$ .

Diagram 1

\begin{aligned} 117^2 &= (110 + 7)^2 \\ &= 110^2 + 2(110)(7) + 7^2 \\ &= 12100 + 1540 + 49 \\ &= 13689 \end{aligned}

Answer

(i) $13689$

(ii) $78^2$

Step 1 · Evaluate $78^2$ using $(a-b)^2$

Express $78$ as $(80 - 2)$ and apply $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 80$ and $b = 2$ .

\begin{aligned} 78^2 &= (80 - 2)^2 \\ &= 80^2 - 2(80)(2) + 2^2 \\ &= 6400 - 320 + 4 \\ &= 6084 \end{aligned}

Answer

(ii) $6084$

(iii) $198^2$

Step 1 · Evaluate $198^2$ using $(a-b)^2$

Express $198$ as $(200 - 2)$ and apply $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 200$ and $b = 2$ .

\begin{aligned} 198^2 &= (200 - 2)^2 \\ &= 200^2 - 2(200)(2) + 2^2 \\ &= 40000 - 800 + 4 \\ &= 39204 \end{aligned}

Answer

(iii) $39204$

(iv) $214^2$

Step 1 · Evaluate $214^2$ using $(a+b)^2$

Express $214$ as $(200 + 14)$ and apply $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 200$ and $b = 14$ .

\begin{aligned} 214^2 &= (200 + 14)^2 \\ &= 200^2 + 2(200)(14) + 14^2 \\ &= 40000 + 5600 + 196 \\ &= 45796 \end{aligned}

Answer

(iv) $45796$

(v) $1104^2$

Step 1 · Evaluate $1104^2$ using $(a+b)^2$

Express $1104$ as $(1100 + 4)$ and apply $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 1100$ and $b = 4$ .

\begin{aligned} 1104^2 &= (1100 + 4)^2 \\ &= 1100^2 + 2(1100)(4) + 4^2 \\ &= 1210000 + 8800 + 16 \\ &= 1218816 \end{aligned}

Answer

(v) $1218816$

(vi) $1120^2$

Step 1 · Evaluate $1120^2$ using $(a+b)^2$

Express $1120$ as $(1100 + 20)$ and apply $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 1100$ and $b = 20$ .

\begin{aligned} 1120^2 &= (1100 + 20)^2 \\ &= 1100^2 + 2(1100)(20) + 20^2 \\ &= 1210000 + 44000 + 400 \\ &= 1254400 \end{aligned}

Answer

(vi) $1254400$

Common Mistakes

Missing the Middle Term: Writing $(a \pm b)^2 = a^2 \pm b^2$ and omitting the $2ab$ term.
Sign Errors in $(a-b)^2$ : Incorrectly writing $(a-b)^2 = a^2 - 2ab - b^2$ instead of $+ b^2$ for the final term.
Inefficient Base Choice: Choosing non-round split values (e.g., splitting $198$ as $190 + 8$ instead of $200 - 2$ ), which makes intermediate calculations more complex.

Question 1

More questions in Exercise 4.3