Question 1: Expand:

i)  ${(x+3)}^2$     ii)  ${ ( 2a+7 ) }^2$     iii)  ${(8+3p)}^2$     iv)  ${(\sqrt{3}x+2)}^2$

v)  ${(4+\sqrt{5}y)}^2$     vi)  ${(6x+11y)}^2$     vii)  $\Big($ $\frac{x}{2}$ $+$ $\frac{y}{3}$ $\Big)^2$     viii)  $\Big($ $\frac{3a}{5}$ $+$ $\frac{5b}{3}$ $\Big)^2$

i)  ${(x+3)}^2 = x^2+6x+9$

ii)  ${ ( 2a+7 ) }^2 = {4a}^2+28a+49$

iii)  ${(8+3p)}^2 = {64+48p+9p}^2$

iv)  ${(\sqrt{3}x+2)}^2 = {3x}^2+4\sqrt{3}x+4$

v)  ${(4+\sqrt{5}y)}^2 = {16+8\sqrt{5}y+5y}^2$

vi)  ${(6x+11y)}^2 = {36x^2+132xy+121y}^2$

vii)  $\Big($ $\frac{x}{2}$ $+$ $\frac{y}{3}$ $\Big)^2 =$ $\frac{x^2}{4}$ $+$ $\frac{1}{3}$ $xy+$ $\frac{y^2}{9}$

viii)  $\Big($ $\frac{3a}{5}$ $+$ $\frac{5b}{3}$ $\Big)^2 =$ $\frac{9a^2}{25}$ $+2ab +$ $\frac{25b^2}{9}$

$\\$

Question 2: Expand:

i)  ${(x-9)}^2$     ii)  ${(6-y)}^2$     iii)  ${(3a-2)}^2$     iv)  ${(8y-5z)}^2$

v)  $\Big($ $\frac{x}{2}$ $-$ $\frac{y}{2}$ $\Big)^2$     vi)  $\Big( 2a-$ $\frac{5}{2}$ $\Big)^2$     vii)  $\Big($ $\frac{2}{a}$ $-$ $\frac{3}{b}$ $\Big)$

i)  ${(x-9)}^2 = x^2-18x+81$

ii)  ${(6-y)}^2 = 36-127+y^2$

iii)  ${(3a-2)}^2 = 9a^2-12a+4$

iv)  ${(8y-5z)}^2 = {64y}^2-80yz+25z^2$

v)  $\Big($ $\frac{x}{2}$ $-$ $\frac{y}{2}$ $\Big)^2 =$ $\frac{x^2}{4}$ $-$ $\frac{1}{2}$ $xy+$ $\frac{y^2}{4}$

vi)  $\Big( 2a-$ $\frac{5}{2}$ $\Big)^2 = {4a}^2-2a+$ $\frac{25}{4}$

vii)  $\Big($ $\frac{2}{a}$ $-$ $\frac{3}{b}$ $\Big)^2 =$ $\frac{4}{a^2}$ $-$ $\frac{12}{ab}$ $+$ $\frac{9}{b^2}$

$\\$

Question 3: Using special expressions to find the value ofː

i)  $({53)}^2$     ii)  $({84)}^2$     iii)  $({1011)}^2$     iv)  $({988)}^2$

i)  $({53)}^2 = ({50+3)}^2 = 2500+300+9 = 2809$

ii)  $({84)}^2 = ({100-16)}^2 = 1000-3200+256 = 7056$

iii)  $({1011)}^2 = ({1000+11)}^2 = 1000000+22000+121 = 1022121$

iv)  $({988)}^2 = ({1000-12)}^2 = 1000000-24000+144 = 976144$

$\\$

Question 4: Using special expressions find the value ofː

i)  $({67)}^2$     ii)  $({795)}^2$     iii)  $({10.9)}^2$     iv)  $({9.2)}^2$

i)  $({67)}^2 = ({70-3)}^2 = 4900-420+9 = 4489$

ii)  $({795)}^2 = ({800-5)}^2 = 640000-8000+25 = 632025$

iii)  $({10.9)}^2 = ({11-0.1)}^2 = 121-2.2+0.01 = 118.81$

iv)  $({9.2)}^2 = ({10-0.8)}^2 = 100-16+0.64 = 84.64$

$\\$

Question 5: If $\Big( x+$ $\frac{1}{x}$ $\Big) = 4$ , find the value of:

i)  $x-$ $\frac{1}{x}$         ii)  $\Big( x^2+$ $\frac{1}{x^2}$ $\Big)$          iii)  $\Big( x^4+$ $\frac{1}{x^4}$ $\Big)$

Given $\Big( x+$ $\frac{1}{x}$ $\Big) = 4$

$\Rightarrow \Big( x+$ $\frac{1}{x}$ $\Big)^2 = 16$

$\Rightarrow x^2+$ $\frac{1}{x^2}$ $+2 = 16$

$\Rightarrow x^2+$ $\frac{1}{x^2}$ $= 14$

Now  $\Big( x-$ $\frac{1}{x}$ $\Big)^2 = x^2+$ $\frac{1}{x^2}$ $-2$

$\Rightarrow \Big( x-$ $\frac{1}{x}$ $\Big)^2 = 14-2 = 12$

Therefore $\Big( x-$ $\frac{1}{x}$ $\Big) = \sqrt{12} = \pm 2\sqrt{3}$

$\Rightarrow \Big( x^4+$ $\frac{1}{x^4}$ $\Big) = \Big( x^2+$ $\frac{1}{x^2}$ $\Big)^2-2 = {14}^2-2 = 194$

$\\$

Question 6: If $\Big( z-$ $\frac{1}{z}$ $\Big) = 6$ find the value of

i)  $\Big( z+$ $\frac{1}{z}$ $\Big)$          ii)  $\Big( z^2+$ $\frac{1}{z^2}$ $\Big)$          iii)  $\Big( z^4+$ $\frac{1}{z^4}$ $\Big)$

Given $z-$ $\frac{1}{z}$ $= 6$

$\Rightarrow z^2+$ $\frac{1}{z^2}$ $-2 = 36$

$\Rightarrow z^2+$ $\frac{1}{z^2}$ $= 38$

$\Rightarrow \Big( z+$ $\frac{1}{z}$ $\Big)^2 = z^2+$ $\frac{1}{z^2}$ $+2 = 38+2 = 40$

$\Rightarrow z+$ $\frac{1}{z}$ $= \pm{}2\sqrt{10}$

$z^4+$ $\frac{1}{z^4}$ $= \Big( z^2+$ $\frac{1}{z^2}$ $\Big)^2-2 = {38}^2-2 = 1442$

$\Rightarrow \Big( z+$ $\frac{1}{z}$ $\Big) = \pm 2\sqrt{10}$

$\Big( z^2+$ $\frac{1}{z^2}$ $\Big) = 38$

$\Big( z^4+$ $\frac{1}{z^4}$ $\Big) = 1442$

$\\$

Question 7: If $\Big( a^2+$ $\frac{1}{a^2}$ $\Big) = 23$ , find the value of $\Big( a+$ $\frac{1}{a}$ $\Big)$

$\Big( a+$ $\frac{1}{a}$ $\Big)^2 = a^2+$ $\frac{1}{a^2}$ $+2$  $= 23+2 = 25$

$\Rightarrow \Big( a+$ $\frac{1}{a}$ $\Big) = \pm 5$

$\\$

Question 8: If $\Big( x^2+$ $\frac{1}{x^2}$ $\Big) = 102$ , find the value of $\Big( x-$ $\frac{1}{x}$ $\Big)$

$\Big( x-$ $\frac{1}{x}$ $\Big)^2 = x^2+$ $\frac{1}{x^2}$ $-2$  $= 102-2 = 100$

$\Rightarrow \Big( x-$ $\frac{1}{x}$ $\Big) = \pm 10$

$\\$

Question 9: If $\Big( 2p+$ $\frac{1}{2p}$ $\Big) = 5$ , find the value of $\Big( 4p^2+$ $\frac{1}{4p^2}$ $\Big)$

$\Big( 2p+$ $\frac{1}{2p}$ $\Big)^2 = 4p^2+$ $\frac{1}{{4p}^2}$ $+2$

$\Rightarrow 4p^2+$ $\frac{1}{4p^2}$ $= 25-2 = 23$

$\\$

Question 10: If $\Big( 3c-$ $\frac{1}{3c}$ $\Big) = 8$ , find the value of $\Big( {9c}^2+$ $\frac{1}{{9c}^2}$ $\Big)$

$\Big( 3c -$ $\frac{1}{3c}$ $\Big)^2 = 9c^2+$ $\frac{1}{9c^2}$ $-2$

$\Rightarrow 9c^2+$ $\frac{1}{9c^2}$ $= 8^2+2 = 64+2 = 66$

$\\$

Question 11: If $( a+b ) = 8$ , and $ab = 15$ , find the value of $a^2+b^2$

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

$\Rightarrow a^2+b^2 = 8^2-2\times{}15 = 64-30 = 34$

$\\$

Question 12: If $a+b = 11$ , and $a^2+b^2 = 61$ , find the value of $ab$

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

$\Rightarrow 2ab = {11}^2-61 = 121-61 = 60$

$\Rightarrow ab = 30$

$\\$

Question 13: If $\ a^2+b^2 = 13$ , and $ab = 6$ , find the value of $( a+b )$

$( a+{b)}^{2\ } = (a^2+b^2+2ab )$ $= 13+12 = 25$

$\Rightarrow ( a+b ) = \pm 5$

$\\$

Question 14: If $a+b = 15$ , and $\ ab = 56$ , find $( a^2+b^2 )$

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

$\Rightarrow a^2+b^2 = {15}^2+2\times 56 = 225-112 = 133$

$\\$

Question 15: If $a-b = 1$  and $ab = 12$ , find $( a^2+b^2 )$

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

$\Rightarrow a^2+b^2 = 1^2+2\times{}12 = 25$

$\\$

Question 16: If $a-b = 5\ \&\ a^2+b^2 = 52\ find\ the\ value\ of\ ab$

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

$\Rightarrow 2ab = a^2+b^2-(a-{b)}^2 = 53-25 = 28$

$\Rightarrow ab = 14$

$\\$

Question 17: If $a^2+b^2 = 52\ \&\ ab = 24$ Find $(a-b)$

${ ( a-b ) }^2 = a^2+b^2-2ab = 52-48 = 4$

$\Rightarrow ( a-b ) = \ \pm 2$

$\\$

Question 18: Find the value of:

(i) $36x^2+49y^2+84xy$ Given $\ x = 3\ \&\ y = 6$

(ii) $25x^2+16y^2-40xy$ Given $\ x = 6\ \&\ y = 7$

(i) $36x^2+49y^2+84xy$  Given $x = 3\ \&\ y = 6$
$\Rightarrow 36x^2+49y^2+84xy = (6x+7y)^2 = (18+42)^2 = 3600$
(ii) $25x^2+16y^2-40xy$ Given $x = 6\ \&\ y = 7$
$\Rightarrow 25x^2+16y^2-40xy = (5x-4y)^2 = (30-28)^2 = 4$