Find the following squares using one of the above identities.

Determine which of these identities will make these calculations easier.

(A) $117^2$

(B) $78^2$

(C) $198^2$

(D) $214^2$

(E) $1104^2$

(F) $1120^2$                                         [Page No. 76]

(A) Use $(a + b)^2$ :

$(117)^2 = (100 + 17)^2$

$= 10000 + 2(100)(17) + 289$

$= 10000 + 3400 + 289$

$= 13689$

(B) Use $(a - b)^2$ :

$(78)^2 = (80 - 2)^2$

$= (80)^2 - 2(80)(2) + (2)^2$

$= 6400 - 320 + 4$

$= 6084$

(C) Use $(a - b)^2$ :

$(198)^2 = (200 - 2)^2$

$= (200)^2 - 2(200)(2) + (2)^2$

$= 40000 - 800 + 4 = 39204$

(D) Use $(a + b)^2$:

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

(E) Use $(a + b)^2$:

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

(F) Use $(a + b)^2$:

$$(1120)^2 = (1100 + 20)^2 = (1100)^2 + 2(1100)(20) + (20)^2 = 1210000 + 44000 + 400 = 1254400$$
Concept Applied
Algebraic identities allow expressions to be expanded or factorised, and values such as $105^2$ or $198^2$ to be computed quickly, by writing a number as $(a \pm b)$.