# Are these the same equations?

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?

• 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. – bames Feb 19 '18 at 5:07
• Wait, how is the one a scaled version? What does that mean? – user532526 Feb 19 '18 at 5:10
• 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 $2$. – anon Feb 19 '18 at 5:11
• What is your definition of same equation? – Tony Ma Feb 19 '18 at 5:30

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.
• @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. – Dave Feb 19 '18 at 5:45
$2x^2+6x+2$ 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 $x^2+3x+1$