Question

1) factorise the following -

i) 7 (2x -y) whole square -25( x - 2 y) + 12

ii) 7 (2x -y) whole square -25 (2x - y ) +12

2) if a=2+root 3 +root 5 and b =3 + root 3 -root 5 , find (a-2) whole square + (b - 3) whole square .

Answer 1

$$\begin{gathered} 7{\left(2\mathrm{x}-\mathrm{y}\right)}^2-25\left(2\mathrm{x}-\mathrm{y}\right)+12 \\ \mathrm{Let}2\mathrm{x}-\mathrm{y}=\mathrm{t} \\ =7{\mathrm{t}}^2-25\mathrm{t}+12 \\ =7{\mathrm{t}}^2-21\mathrm{t}-4\mathrm{t}+12 \\ =7\mathrm{t}\left(\mathrm{t}-3\right)-4\left(\mathrm{t}-3\right) \\ =\left(\mathrm{t}-3\right)\left(7\mathrm{t}-4\right) \\ \mathrm{Putting}2\mathrm{x}-\mathrm{y}=\mathrm{t}\mathrm{we}\mathrm{get}, \\ =\left(2\mathrm{x}-\mathrm{y}-3\right)\left(14\mathrm{x}-7\mathrm{y}-3\right) \end{gathered}$$

$$\begin{gathered} 2)\mathrm{a}=2+\sqrt{3}+\sqrt{5} \\ \mathrm{b}=3+\sqrt{3}-\sqrt{5} \\ \mathrm{Now}{\left(\mathrm{a}-2\right)}^2={\left(2+\sqrt{3}+\sqrt{5}-2\right)}^2 \\ ={\left(\sqrt{3}+\sqrt{5}\right)}^2=3+5+2\sqrt{15}=8+2\sqrt{15} \\ \mathrm{b}-3=\sqrt{3}-\sqrt{5} \\ {\left(\mathrm{b}-3\right)}^2={\left(\sqrt{3}-\sqrt{5}\right)}^2 \\ =3+5-2\sqrt{15}=8-2\sqrt{15} \\ \mathrm{So}{\left(\mathrm{a}-2\right)}^2+{\left(\mathrm{b}-3\right)}^2=8+2\sqrt{15}+8-2\sqrt{15} \\ =16 \end{gathered}$$