In my textbook the formula of an ellipse is noted as $$b^2 x^2 + a^2 y^2 = a^2 b^2$$
Now I have a formula that goes $$x^2 + 4y^2 = 17$$
How is this formula valid, as $1\cdot 4$ does not equal $17$?
Edit: I just realized that if: $$nx + my = nm$$ and $$2x + 4y=8$$ and multiply the equation by 2 to get $$4x + 8y = 16$$ while this is true: $$2*4=8$$ this doesn't work anymore because we multiplied n and m separately on the left side and together on the right: $$4*8 = 16$$ I literally suck and Im out of here...
Let divide by $17$ to obtain
$$x^2 + 4y^2 = 17 \iff \frac{x^2}{17} + \frac{4y^2}{17} = 1$$
then note that $$\frac{4}{17}=\frac{1}{\frac{17}4}$$
then
$$\frac{x^2}{17} + \frac{4y^2}{17} = 1 \iff \frac{x^2}{17} + \frac{y^2}{\frac{17}4} = 1 \iff \frac{x^2}{(\sqrt{17})^2} + \frac{y^2}{\left(\frac{\sqrt{17}}2\right)^2} = 1$$
which is the form you are looking for with $a=\sqrt{17}$ and $b=\frac{\sqrt{17}}2$, indeed as noticed in the comments we have that
$$b^2x^2 + a^2y^2 = a^2b^2 \iff \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
are equivalent equations for an ellipse centered at origin.