Q.1. Expand;

1. ${(x+3)}^2=x^2+6x+9$
2. ${\left(2a+7\right)}^2={4a}^2+28a+49$
3. ${(8+3p)}^2={64+48p+9p}^2$
4. ${(\sqrt{3}x+2)}^2={3x}^2+4\sqrt{3}x+4$
5. ${(4+\sqrt{5}y)}^2={16+8\sqrt{5}y+5y}^2$
6. ${(6x+11y)}^2={36x^2+132xy+121y}^2$
7. ${(\frac{x}{2}+\frac{y}{3})}^2={\frac{x^2}{4}+\frac{1}{3}xy+\frac{y}{9}}^2$
8. ${(\frac{3a}{5}+\frac{5b}{3})}^2=\frac{9a^2}{25}+2ab+\frac{25b^2}{9}$

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Q.2. Expand

1. ${(x-9)}^2=x^2-18x+81$
2. ${(6-y)}^2=36-127+y^2$
3. ${(3a-2)}^2=9a^2-12a+4$
4. ${(8y-5z)}^2={64y}^2-80yz+25z^2$
5. ${\left(\frac{x}{2}-\frac{y}{2}\right)}^2={\frac{x}{4}}^2-\frac{1}{2}xy+\frac{y^2}{4}$
6. ${\left(2a-\frac{5}{2}\right)}^2={4a}^2-2a+\frac{25}{4}$
7. $\left(\frac{2}{a}-\frac{3}{b}\right)=\frac{4}{a^2}-\frac{12}{ab}+\frac{9}{b^2}$

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Q.3. Using special expressions to find the value ofː

1. $({53)}^2=({50+3)}^2=2500+300+9=2809$
2. $({84)}^2=({100-16)}^2=1000-3200+256=7056$
3. $({1011)}^2=({1000+11)}^2=1000000+22000+121=1022121$
4. $({988)}^2=({1000-12)}^2=1000000-24000+144=976144$

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Q.4. Using special expressions find the value ofː

1. $({67)}^2=({70-3)}^2=4900-420+9=4489$
2. $({795)}^2=({800-5)}^2=640000-8000+25=632025$
3. $({10.9)}^2=({11-0.1)}^2=121-2.2+0.01=118.81$
4. $({9.2)}^2=({10-0.8)}^2=100-16+0.64=84.64$

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Q.5. If  $\ \ \left(x+\frac{1}{x}\right)=4$, find the value of:

1. $x-\frac{1}{x}$
2. $\left(x^2+\frac{1}{x^2}\right)$
3. $\left(x^4+\frac{1}{x^4}\right)$

$\left(x+\frac{1}{x}\right)=4$

${\left(x+\frac{1}{x}\right)}^2=16$

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

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

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

${\left(x-\frac{1}{x}\right)}^2=14-2=12$

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

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

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Q.6. If $\left(z-\frac{1}{z}\right)=6$ find the value of

(i) $\left(z+\frac{1}{z}\right)$

(ii) $\left(z^2+\frac{1}{z^2}\right)$

(iii) $\left(z^4+\frac{1}{z^4}\right)$

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

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

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

${\left(z+\frac{1}{z}\right)}^2=z^2+\frac{1}{z^2}+2 =38+2=40$

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

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

$\left(z+\frac{1}{z}\right)=\pm{}2\sqrt{10}$

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

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

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Q.7. If  $\ \ \left(a^2+\frac{1}{a^2}\right)=23$, find the value of $\left(a+\frac{1}{a}\right)$

${\left(a+\frac{1}{a}\right)}^2=a^2+\frac{1}{a^2}+2$

$=23+2=25$

$\left(a+\frac{1}{a}\right)=\pm{}5$

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Q.8.If  $\ \ \left(x^2+\frac{1}{x^2}\right)=102$, find the value of $\left(x-\frac{1}{x}\right)$

${\left(x-\frac{1}{x}\right)}^2=x^2+\frac{1}{x^2}-2$

$= 102-2=100$

Therefore $\ \left(x-\frac{1}{x}\right)=\pm{}10$

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Q.9. If  $\ \ \left(2p+\frac{1}{2p}\right)=5$, find the value of $\left(4p^2+\frac{1}{4p^2}\right)$

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

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

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Q.10. If $\ \ \left(3c-\frac{1}{3c}\right)=8$, find the value of $\left({9c}^2+\frac{1}{{9c}^2}\right)$

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

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

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Q.11. If $\ \ \left(a+b\right)=8$, and $ab=15$, find the value of $a^2+b^2$

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

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

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Q.12. If  $\ \ a+b=11$, and $a^2+b^2=61$, find the value of $ab$

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

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

$ab=30$

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Q.13. If $\ a^2+b^2=13$, and $ab=6$, find the value of $\left(a+b\right)$

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

$=13+12=25$

$Hence\ \left(a+b\right)=\pm{}5$

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Q.14. If $a+b=15$, and $\ ab=56$, find $\left(a^2+b^2\right)$

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

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

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Q.15. If $a-b=1\ and\ ab=12,\ find\ \left(\ a^2+b^2\right)$

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

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

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

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

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

$or\ \ ab=14$

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

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

$\left(a-b\right)=\ \pm{}\ 2$

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Q.18 Find the value of:

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

$36x^2+49y^2+84xy=(6x+7y)^2=(18+42)^2=3600$

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

$25x^2+16y^2-40xy=(5x-4y)^2=(30-28)^2=4$