Trigo vs geometry

$\text{Let}\space x=\frac{a+\sqrt{b}}{2}$

$\text{By the cosine rule,}$

$\begin{aligned} 6^2&={(x-4)}^2+{(x-5)}^2-2(x-4)(x-5)\cos(60°)\\ 36&=(x^2-8x+16)+(x^2-10x+25)-2(x^2-9x+20)\left(\frac12\right)\\ 36&=x^2-8x+16+x^2-10x+25+x^2+9x-20\\ 0&=x^2-9x-15\\ x&=\frac{-(-9)±\sqrt{{(-9)}^2-4(1)(-15)}}{2(1)}\\ x&=\frac{9±\sqrt{141}}{2} \end{aligned}$

$\text{Since}\space \frac{a+\sqrt{b}}{2}>0,$

$\begin{aligned} \frac{a+\sqrt{b}}{2}&\iff\frac{9+\sqrt{141}}{2}\\ a+\sqrt{b}&\iff 9+\sqrt{141} \end{aligned}$

$\Rightarrow a=9,\space b=141$

$\text{Therefore,}$

$\begin{aligned} 20a-b&=20(9)-141\\ &=180-141\\ 20a-b&=\boxed{39} \end{aligned}$

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