# Find when equation has EXACTLY one solution [closed]

Simple idea, but I cannot figure out a way to do this. What I want is to find for which value of $$k$$ some equation $$f(x,k)=g(x)$$ has exactly one real solution.

For example, say $$f(x,k)=x^2+k$$ and $$g(x)=6x^2+3$$. Looking at this particular example, it's not too hard to figure out the value of $$k$$ we want is $$3$$. However, it's much more difficult for more complex functions, like $$f(x,k)=kx^4-x^3$$ and $$g(x)=\ln(x)-x^2$$. Looking at the graphs of these functions, it's easy to see that there must be exactly one value of $$k$$ where they are equal exactly once. Finding said value, however, eludes me.

The nature of this problem leaves it begging to be solved using limits, and calculus is never too far behind when limits are involved, but I'm not too familiar with multivariable calculus...

## closed as too broad by Luca Bressan, Dietrich Burde, Matthew Daly, N. F. Taussig, Lee David Chung LinSep 12 at 1:52

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• The answer strongly depends on the forms of $f$ and $g$ – caverac Sep 11 at 18:16
• Some equation? This is too broad. Most equations will not work, e.g., $f(x,k)=(x^2+1)^k$. – Dietrich Burde Sep 11 at 18:42