Question 1: Expand:

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

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

i) $\displaystyle {(x+3)}^2 = x^2+6x+9$

ii) $\displaystyle { ( 2a+7 ) }^2 = {4a}^2+28a+49$

iii) $\displaystyle {(8+3p)}^2 = {64+48p+9p}^2$

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

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

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

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

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

$\displaystyle \\$

Question 2: Expand:

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

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

i) $\displaystyle {(x-9)}^2 = x^2-18x+81$

ii) $\displaystyle {(6-y)}^2 = 36-127+y^2$

iii) $\displaystyle {(3a-2)}^2 = 9a^2-12a+4$

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

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

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

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

$\displaystyle \\$

Question 3: Using special expressions to find the value ofː

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

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

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

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

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

$\displaystyle \\$

Question 4: Using special expressions find the value ofː

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \Rightarrow ab = 30$

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \\$

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

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

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

$\displaystyle \Rightarrow ab = 14$

$\displaystyle \\$

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

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

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

$\displaystyle \\$

Question 18: Find the value of:

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

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

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