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February 22nd, 2016, 12:09 AM   #1
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Question 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
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February 22nd, 2016, 12:37 AM   #2
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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
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February 22nd, 2016, 01:00 AM   #3
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Math Focus: tetration
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Originally Posted by manus View Post
x^2 - 2x - 3 = 0 true
-x^2 + 2x + 3 = 0 true
x^2 - 2x -3 = -x^2 + 2x + 3 is a false generalisation
x^2 - 2x - 3 = -x^2 + 2x + 3 is only true for the zeroes of the polynomial i.e x= -1 or x = 3
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February 22nd, 2016, 10:33 AM   #4
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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 x-axis at (-1, 0) and (3, 0). There can be many different parabola that cross the x-axis at exactly the same place!
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