If a bilinear form is non-singular, is the associated quadratic form also non singular?

Question

If $f$ is a non-singular bilinear form on a finite-dimensional vector field over a field of characteristic $\neq 2$, is the associated quadratic form also non-singular? What can be said if $f$ is singular?

Here are how the terms have been defined for us -

1. The quadratic form $Q_f$ associated with a bilinear form $f$ is defined by $Q_f(x) = f(x, x)$.
2. The symmetric bilinear form $f_Q$ associated with a quadratic form $Q$ is defined by $f_Q(x, y) = Q(x+y) - Q(x) - Q(y)$.
3. A bilinear form $f$ is non-singular if $L_f(x) = f(x, \_)$ and $R_f(x) = f(\_,x)$ are both injective.
4. A quadratic form is non-singular if the associated bilinear form is non-singular.

My guess is that the statement is true, so I tried proving $L_{f_Q}$ and $R_{f_Q}$ are injective but couldn't.