I encounter that quadratic inequality can have two ways of graphing, line(s) with the area of solutions and parts of a parabola. How is that happen?

Example: x²-4x ≥ 14. When I try to graph it on graph calculators, it is either represented as lines on x=-2.243 and x=6.243 with the rest of its solutions or a parabola with the ranges only above the x-axis (y>0). Why there are two versions of the graph? Please help.

  • Erdos Erdos
    0

    Your question is very unclear. I am not sure what you are actually asking about. Please write your question more carefully. It might be a good idea to include an example to make it even more clear.

    • Ken001 Ken001
      0

      Oh thank you for telling me

1 Answer

This is similar to your previous question.

If you are looking for $x$ such that 
\[x^2-4x\geq 14,\]
then 
\[x^2-4x- 14\geq 0.\]
Finding the roots using the quadratic formula, you get
\[x_1,x_2=\frac{+4\pm\sqrt{14+56}}{2}=\frac{+4\pm\sqrt{70}}{2}.\]
Hence the inequality is satisfied if 
\[ x\geq \frac{+4+\sqrt{70}}{2}   \text{or}    x\leq \frac{+4-\sqrt{70}}{2},\]
which are two line segments if graphed on $x$-axis. 

However, if you view this problem as the graph of the function $y=x^2-4x- 14$, then you want the portion of the graph that lies above $x$-axis, i.e. $y=x^2-4x- 14\geq 0.$

So it is the same problem, viewed from two different points of view. The first view is one dimensional where we only have the variable $x$, while in the second case we view the problem in two dimensions and have two variables $x,y$.

Erdos Erdos
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  • Erdos Erdos
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    If you find my solutions helpful, I would appreciate a tip!

  • Ken001 Ken001
    +1

    Thank you very much for your answer, this is so helpful. But, I’m really sorry that I can’t give you any tips since I don’t have a bank account yet. Sorry and thank you very much!

    • Erdos Erdos
      0

      It's alright. I am glad I was able to help.

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