$\large (x-100)^4+(x-102)^4+30(x-100)^2(x-102)^2$ If $x$ is a real number larger than 101, then find the minimum value of the expression above.

Let this value be denoted as $y$ , submit your answer as $y$ plus the value of $x$ at this minimum value.

Give your answer to 2 decimal places.

The answer is 115.86.

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Set $t=x-101 ; t>0$ , the expression becomes $(t+1)^4+(t-1)^4+30(t-1)^2(t+1)^2$ $= [(t+1)^2-(t-1)^2]^2+32(t^2-1)^2$ $=32t^4-48t^2+32$ $=32\bigg(t^2-\frac{3}{4}\bigg)^2+14\geq 14$ So the minimum is $14$ and the equality holds when $x=101+\frac{\sqrt{3}}{2}$ $\Rightarrow 14+101+\frac{\sqrt{3}}{2}\approx 115.86$