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February 21st, 2016, 11:09 PM  #1 
Newbie Joined: Feb 2016 From: Shoeburyness, England Posts: 1 Thanks: 0  Quadratic equations and negative values of x^2
Hi, I'm completely new to this forum so please bear with me if this question has been asked before. I've been going back to basics and reinforcing a lot of my maths knowledge and my current topic is quadratic equations and sketching graphs. I've got no problems with this at all  my question is more of a conceptual issue. Take the example x^2  2x 3 = 0 Bog standard quadratic with a U shaped graph with solutions for x = 1 and 3 and the largest value of y coords = (1,4). No probs. If I then reverse the signs on that equation: x^2 +2x +3 = 0 Again in terms of the math I totally get that the x solutions remain unchanged and producing values for y coords produces (1,4) The problem I'm having is that conceptually to me the two equations are the same  I've just rearranged the signs, so x^2 2x 3 = x^2 +2x +3 = 0 So if they are they same, why does changing the signs produce a mirrored image of the quadratic curve about the x axis? Is there a mathematical explanation for this? Am I right to say that the two equations are equal when they produce different graphs? I hope my question makes sense and I look forward to hearing from the forum. Regards Mark 
February 21st, 2016, 11:37 PM  #2 
Senior Member Joined: Dec 2015 From: holland Posts: 162 Thanks: 37 Math Focus: tetration 
x^2  2x  3 = 0 true x^2 + 2x + 3 = 0 true x^2  2x 3 = x^2 + 2x + 3 is a false generalisation 
February 22nd, 2016, 12:00 AM  #3 
Senior Member Joined: Dec 2015 From: holland Posts: 162 Thanks: 37 Math Focus: tetration  
February 22nd, 2016, 09:33 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Yes, the graph of $\displaystyle y= x^2 2x 3= (x 3)(x+ 1)$ is a parabola, opening upward, with vertex at (1, 4). The graph of $\displaystyle y= (x^2 2x 3)= x^2+ 2x+ 3$ is a parabola, opening downward ,with vertex at (1, 4). Both of them cross the xaxis at (1, 0) and (3, 0). There can be many different parabola that cross the xaxis at exactly the same place!


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