How can I solve this simple equation?

Questions
,
1

After simplification, I have equation express like below,

Solve[Log[Sqrt[(1 + x + b*x^2)]] == (1 + x + a*x^2), x]

There a and b is known variable and I want to solve x as a function of a and b.

The mathematic always shown the error: “This system cannot be solved with the methods available to Solve.”
How can I solve it?
Thank you a lot!

2

There is obviously no closed-form solution. Need to solve numerically after assigning values to a and b, for example

In[10]:= a = 2; b = 3; FindRoot[
 Log[Sqrt[(1 + x + b*x^2)]] == (1 + x + a*x^2), {x, 1}]

During evaluation of In[10]:= FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>

Out[10]= {x -> -0.422306}
3

I would not describe “This system cannot be solved with the methods available to Solve.” as an error message.