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Question Number 211658 by efronzo1 last updated on 15/Sep/24

$$\underbrace{}$$

Answered by mehdee1342 last updated on 15/Sep/24

$$y=\sqrt{19-8\sqrt{3}}\Rightarrow y^2=19-8\sqrt{3}$$$$y^4-18y^2+361=192$$$$\stackrel{x=y^2}{\Rightarrow }{x}^2-18x+169=0\Rightarrow a=18\mathrm{\&}b=361$$$$\Rightarrow a+b=379✓$$$$$$

Commented by Frix last updated on 15/Sep/24

$$x^2+18x+169=0\Rightarrow x=9\pm 2\sqrt{22}\mathrm{i}\ne \sqrt{19-8\sqrt{3}}$$

Answered by Frix last updated on 15/Sep/24

$$\sqrt{19-8\sqrt{3}}=4-\sqrt{3}$$$$$$$$1.$$$$x_1=4-\sqrt{3}$$$$a,b\in \mathbb{Q}\Rightarrow x_2=4+\sqrt{3}$$$$(x-x_1)(x-x_2)=x^2-8x+13$$$$a=8\wedge b=13$$$$a+b=21$$$$$$$$2.$$$$x^2-ax+b=0\Rightarrow x=\frac{a}{2}\pm \frac{\sqrt{a^2-4b}}{2}$$$$\Rightarrow \frac{a}{2}=4\wedge \frac{\sqrt{a^2-4b}}{2}=\sqrt{3}$$$$\Rightarrow a=8\wedge b=13$$$$a+b=21$$

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