Finding the common of tangency

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There're $2$ curves: $y=x^3-2x+1$ and $y=x^2+2ax+1$.

They passes through the same point and have a common tangent at that point.
And I'd like to find the common point of tangency and the value of $a$.

My attempts:

Let $f(x)=x^3-2x+1$ and $g(x)=x^2+2ax+1$ I have tried to use $x=k$ as the common point. Therefore the tangent is $(k, k^2-2k+1)$.

But I don't know the steps afterwards. Please help me to solve this question

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1 Answer

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Hint. You want to solve $$ x^3-2x+1=x^2+2ax+1 \quad (\text{same point}) $$ and $$ 3x^2-2=2x+2a \quad (\text{common tangent at that point}). $$

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