Ways to check positive definiteness of bilinear form

Question

For each real number $\alpha$, we define the bilinear form $F_{\alpha}:\mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R} $ by $\displaystyle F_{\alpha}(((x_1, x_2, x_3), (y_1, y_2, y_3)) = 2x_1y_1 + (\alpha + 5)x_1y_2 + x_1y_3 + (\alpha + 5)x_2y_1 -(2\alpha + 4)x_2y_2 + 2x_2y_3 + x_3y_1 + 2x_3y_2 + 2x_3y_3.$ Find the set of $\alpha \in \mathbb{R}$ such that $F_{\alpha}$ is positive definite.

Is there any other way to check positive definiteness instead of extracting the underlying symmetric matrix and the eigenvalues ? Can the same be applied for quadratic forms ?