For a real constant $k$, consider $$\mathrm{f}(x)=x^2-2kx+k+2.$$ Complete the square in terms of \

Question

For a real constant $k$, consider $$\mathrm{f}(x)=x^2-2kx+k+2.$$ Complete the square in terms of $k$. Find all values of $k$ for which $\mathrm{f}(x)>0$ for every real $x$. For which values of $k$ does the graph touch the $x$-axis? State each point of contact.

Accepted Answer

(a)$\mathrm{f}(x)=(x-k)^2-k^2+k+2$; (b) Positive for every real $x$ when $-1<k<2$; (c) Contact at $(-1,0)$ for $k=-1$, and $(2,0)$ for $k=2$.

Quadratic Functions, Equations and Inequalities

Quadratic Functions, Equations and Inequalities

Solution:

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