Stuck on finding the range of one unknown in a quadratic equation with 2 unknowns

algebra-precalculus

For context this is the question that I am stuck on: Find the range of values of a for which $x^2 + ax + 2(a – 1)$ is always greater than 1.

I have tried to use the discriminant $b^2-4ac$ but to no avail, I am still unable to find the range of values of a as I have two unknowns, x and a in this case. I also tried to substitute different values of x into the equation but realised that it did not really make sense and as I got different values of a each time.

May I know how I can solve this question?


Hint: you need $$x^2+ax+2a-2>1$$ or $$x^2+ax+2a-3>0$$ or $$\left(x+\frac{a}{2}\right)^2+2a-\frac{a^2}{4}-3>0$$
Note that $\left(x+\frac{a}{2}\right)^2\ge0$.

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