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

a=2+√(5) b=1a b=1(2 √(5))(2+√(5)(2 √(5))=2 √(5)4 5 b=2 √(5) .So a^2+b^2 ⇒ (2+√(5))^2+(2 √(5))^2 ⇒ 4+5+4√(5)+4+5 4√(5) ⇒ ...

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Solving a Quadratic Equation by Completion of Squares Method

If a=2+√ 5 ...

Question

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

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a = 2 + \sqrt{5}

b = \frac{1}{a}

a^{2} + b^{2}

a + b = 5

a b = 6

a^{2} + b^{2}

a + b = 1

a - b = 7

a^{2} + 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

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}}

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