Satisfying Conditions for Inner Product in Space $R^2$

Question

The solution states that this is an inner product iff $b=c$ and $d>b^2$. I am not sure how to solve/approach this problem. I know that there are 4 conditions for the inner product condition to satisfy. Can you explain how to use those four properties to find conditions for b,c, and d?

P.S. Properties of inner product

1. $<x,y>=<y,x>$
2. $<x+z,y>=<x,y>+<z,y>$
3. $<cx,y>=c<x,y>=<x,cy>$
4. $<x,x>=0$ if $x\neq0 $

Answer 1

Eventually, you learn that positive definiteness will require that the determinant of the matrix of coefficients be positive. But you can see it directly by completing the square: With $x=(x_1,x_2)$, $$\langle x,x \rangle = x_1^2 + 2bx_1x_2 + dx_2^2 = (x_1+bx_2)^2 + (d-b^2)x_2^2.$$ Can you finish now?