Let $A$ be a two-dimensional unitary commutative algebra over $\mathbb{R}$ with identity $u$. I have to show that $A$ has a basis $u, v$ such that $v^2=ru$ for some $r\in\mathbb{R}$.
My attempt so far is to write $v=\alpha u + \beta w$ for some $\alpha,\beta\in\mathbb{R}$ where $u, w$ is another basis. Then $$v^2=(\alpha u + \beta w)(\alpha u + \beta w) = \alpha^2u+\beta^2w^2+2\alpha\beta w = \alpha^2u+\beta^2(\gamma u+\delta w) + 2\alpha\beta w =(\alpha^2+\beta^2\gamma)u+(2\alpha\beta+\beta^2\delta)w,$$ where I have written $w^2$ as $\gamma u+\delta w$ for some $\gamma, \delta\in\mathbb{R}$.
I then let $r=\alpha^2+2\alpha\beta$ and try to conclude (somehow) that $2\alpha\beta=0$.
However, I'm not very confident that this approach is correct or even achieves what is being asked. In particular, I don't think it shows that $u,v$ is a basis.
Any help would be greatly appreciated! Thanks in advance