Equation with variable $λ$ that should be $R$ for $x$

Question

I need to find for $λ\in R$ the domain of $f(x)=\sqrt{((λ-2)x^2-2λx+2λ-3)}$

It should be $λ\in[6,+∞ )$ as per my book but I dont understand why. Sorry for my english

Answer 1

The domain of the $f(x)$ is the set of values of $x$ which satisfy the following inequation: $$(λ-2)x^2-2λx+2λ-3\gt0\tag{i}$$ For the domain to be the set of real numbers, (i) must hold for any real $x$. Since the expression is a quadratic in $x$, we know that it will be always positive if its discriminant is non-positive, and the leading coefficient positive. This gives us the following conditions on $\lambda$:- $$\lambda-2\gt 0\tag{ii}$$ $$4\lambda^2-4(2\lambda-3)(\lambda-2)\le 0\tag{iii}$$ Solving (ii) and (iii) the asnwer will be $\lambda\in[6,\infty)$