Evaluation of expressions

If a²-3a+1=0, find

(i)a²+1/a²
(ii) a³+1/a³

Correct result:

x₁ =  7
x₂ =  7
y₁ =  18
y₂ =  18

Solution:

$$\begin{gathered} a^2-3a+1=0 \\ {a}^2-3a+1=0 \\ {a}^2-3a+1=0 \\ p=1;q=-3;r=1 \\ D=q^2-4pr=3^2-4\cdot 1\cdot 1=5 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{3\pm \sqrt{5}}{2}} \\ {a}_{1,2}=1.5\pm 1.11803398875 \\ {a}_1=2.61803398875 \\ {a}_2=0.38196601125 \\ \text{ Factored form of the equation: } \\ (a-2.61803398875)(a-0.38196601125)=0 \\ {x}_1=a_1^2+1\mathrm{/}{a}_1^2=2.618^2+1\mathrm{/}2.618^2=7 \end{gathered}$$

Our quadratic equation calculator calculates it.

$x_2=a_2^2+1\mathrm{/}{a}_2^2=0.382^2+1\mathrm{/}0.382^2=7$

$y_1=a_1^3+1\mathrm{/}{a}_1^3=2.618^3+1\mathrm{/}2.618^3=18$

$y_2=a_2^3+1\mathrm{/}{a}_2^3=0.382^3+1\mathrm{/}0.382^3=18$

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