If a-2- 5 and b- 1 - a find a 2 - b 2-

a=2+√(5) , b= 12+√(5) a^2+b^2 = (2+√(5))^2 + 1(2+√(5))^2 #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; ...

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Factorisation by Common Factors

If a=2+√ 5 ...

Question

If $a = 2 + \sqrt{5}$ and $b = \frac{1}{a}$ find $a^{2} + b^{2}$ .

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Solution

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a = 2 + \sqrt{5}

b = \frac{1}{a}

a^{2} + b^{2}

(a + b) : (a - b) = 1 : 5

(a^{2} - b^{2}) : (a^{2} + b^{2}) .

a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}

b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}

\frac{a^{2} + a b + b^{2}}{a^{2} - a b + b^{2}}

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{[1 - {(\frac{a}{b})}^{- 2}] a^{2}}{(\sqrt{a} - \sqrt{b})^{2} + 2 \sqrt{a b}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 1

a + b = 5,

a

b

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{[1 - {(\frac{a}{b})}^{- 2}] a^{2}}{{(\sqrt{a} - \sqrt{b})}^{2} + 2 \sqrt{a b}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ =

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