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If I have $2x^2+6x+2=0$ and $x^2+3x+1=0$ and then I multiply both sides of $x^2+3x+1=0$ by 2, I get $2x^2+6x+2=0$, which is the same as the first equation. However, when I graph $2x^2+6x+2$ and $x^2+3x+1$, they aren't the same graphs. Why is that?

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    $\begingroup$ One is a scaled version of the other. Your observation that $2(x^2+3x+1) = 0$ is the same as $2x^2 + 6x + 2 = 0$ tells you that $x^2 + 3x+1$ and $2x^2 + 6x + 2$ have the same roots. $\endgroup$ – bames Feb 19 '18 at 5:07
  • $\begingroup$ Wait, how is the one a scaled version? What does that mean? $\endgroup$ – user532526 Feb 19 '18 at 5:10
  • $\begingroup$ It means one is the other multiplied by some number. The two equations with $=0$ are logically equivalent, the two equations with $y=$ are not equivalent. Graphically, the graph of the scaled quadratic will be the same graph as the former, but stretched away from the $x$-axis by a factor of $2$. $\endgroup$ – anon Feb 19 '18 at 5:11
  • $\begingroup$ What is your definition of same equation? $\endgroup$ – Tony Ma Feb 19 '18 at 5:30
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It is true that $2x^2+6x+2=0$ and $x^2+3x+1=0$ are same equations but when you are plotting you are plotting the functions $y(x) =2x^2+6x+2$ and $y(x)=x^2+3x+1$ which are different functions. One is a scaled version of the other.

To have same roots you need to have the curves intersect $y=0$ at the same points, which these two curves would do.

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    $\begingroup$ They are not the same equations. $\endgroup$ – user532449 Feb 19 '18 at 5:15
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    $\begingroup$ @YoloInver these equations $2x^2+6x+2=0$ and $x^2+3x+1=0$ are indeed the same equations. However, the functions $f(x)=2x^2+6x+2$ and $g(x)=x^2+3x+1$ are not the same. $\endgroup$ – Dave Feb 19 '18 at 5:45
  • $\begingroup$ @YoloInver You’re right, they are different equations but share the same solution set. $\endgroup$ – Michael Hoppe Feb 19 '18 at 7:24
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$2x^2+6x+2$ and $x^2+3x+1$ have the same roots, this does not mean that they are the same equations.

$2x^2+6x+2\ne x^2+3x+1$, since $2x^2+6x+2=2(x^2+3x+1)$, it is vertically stretched twice as much as $x^2+3x+1$

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