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What is the value of b+c if x^2+bx+c>0 only when x\in (-\infty, 2)\cup(2,\infty)?

Oct 19, 2022

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What is the value of b+c if x^2+bx+c>0 only when $$x\in (-\infty, 2)\cup(2,\infty)?$$?

$$x^2+bx+c>0$$

this is a concave up parabola.

If this function is greater than zero for all values of x except for x=2  then the vertex is  (2,0)

And there will only be one root.

so

$$x^2+bx+c=(x-2)^2\\ b=-4,\;c=4\\ b+c=0$$

Oct 20, 2022