Question

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Answer 1

E(x)=[5/(x-2)+2/(x+2)-6/(x²-4]:[(x²+4)/(x²-4)+1]

b) x=1/(√5-1)-1/(√5+1)=(√5+1)/(5-1)-(√5-1)/(5-1)=

(√5-1-√5+1)/4=0

E(x)=[5/(0-2)+2/(0+2)-6/(0-4]:[(0+4)/(0-4)+1]=

(-5/2+2/2+6/4):(4/(-4+1)=

[(-5+2+3)/2]:(-3)=

0/2:(-3)=0

c) E(a)=(1/2)a+3

E(x)=[5/(x-2)+2/(x+2)-6/(x²-4]:[(x²+4)/(x²-4)+1]

=[5(x+2)+2(x-2)-6]/(x²-4):[(x²+4+x²+4)/(x²-4)]=

=[(7x+0)/(x²-4)]:[(x²+4+x²-4)/(x²-4)]

=7x/(x²-4)×(x²-4)/2x²

=(7x)/2x²=7/(2x)

E(a)=(1/2)a+3=7/(2a)

a/2-7/(2a)+3=0

a²-7+6a=0

a²+6a-7=0

a²-a+7a-7=0

a(a-1)+7(a-1)=0

(a-1)(a+7)=0

a1=1

a2=-7

Answer 2

$b)~E(x)=\Bigg(\cfrac{5}{x-2} +\cfrac{2}{x+2} -\cfrac{6}{x^2-4}\Bigg):\Bigg(\cfrac{x^2+4}{x^2-4} +1\Bigg)\\\\ E(x)=\cfrac{5(x+2)+2(x-2)-6}{x^2-4} :\cfrac{x^2+4+x^2-4}{x^2-4} \\\\E(x)=\cfrac{5x+10+2x-4-6}{x^2-4} \cdot \cfrac{x^2-4}{2x^2} \\\\E(x)=\cfrac{7x}{2x^2} \Rightarrow E(x)=\cfrac{7}{2x} \\\\x=\cfrac{1}{\sqrt{5}-1 } -\cfrac{1}{\sqrt{5}+1 } \Rightarrow E(x)=\cfrac{7}{2\Bigg(\cfrac{1}{\sqrt{5}-1 }-\cfrac{1}{\sqrt{5} +1}\Bigg) } \Rightarrow$

$\Rightarrow E(x)=\cfrac{7}{2 \cdot \cfrac{\sqrt{5}+1 -\Big(\sqrt{5}-1\Big) }{\Big(\sqrt{5} -1\Big)\Big(\sqrt{5}+1\Big) } } \Rightarrow E(x)=\cfrac{7}{2\cdot\cfrac{\sqrt{5}+1-\sqrt{5}+1 }{\sqrt{5}\cdot \sqrt{5} +\sqrt{5}-\sqrt{5} -1} } \Rightarrow \\\\\Rightarrow E(x)=\cfrac{7}{2 \cdot \cfrac{2}{4} } \Rightarrow E(x)=\cfrac{7}{\cfrac{4}{4} } \Rightarrow \boxed{E(x)=7}$

$c)~E(a)=\cfrac{1}{2} a+3~~~~~~~~~~~~~~~~~~~~~~E(a)=\cfrac{7}{2a} \\\\E(a)=\cfrac{1}{2}a+3\Rightarrow \cfrac{7}{2a}=\cfrac{1}{2}a+3 \Rightarrow 7=a^2+6a \Rightarrow -a^2-6a+7=0 \Big|\cdot (-1) \Rightarrow \\\\\Rightarrow a^2+6a-7=0\\\\ \Delta=6^2-4 \cdot 1 \cdot (-7)=36+28=64 > 0\\\\a_1=\cfrac{-6-\sqrt{64} }{2 \cdot 1} =\cfrac{-6-8}{2}= \cfrac{-14}{2} =-7~~~~~~~~~~~~~~~~\boxed{a_1=-7}\\\\a_2=\cfrac{-6+\sqrt{64} }{2 \cdot 1}=\cfrac{-6+8}{2} =\cfrac{2}{2}=1 ~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{a_2=1}$