The roots of the quadratic equation are

$$z_{1,2}=-\frac{a_1}{2}\pm\sqrt{\frac{a_1^2}{4}-a_2}\tag{1}$$

For $a_1^2/4\ge a_2$, the roots are real-valued. In that case we require

$$-1<z_{1,2}<1\tag{2}$$

Let's start with the first inequality (from the left) in $(2)$:

$$-1+\frac{a_1}{2}<\pm\sqrt{\frac{a_1^2}{4}-a_2}\tag{3}$$

Squaring $(3)$ and rearranging gives

$$a_1-a_2<1\tag{4}$$

In the same way you can obtain the other inequality by considering the second inequality in $(3)$:

$$a_1+a_2>-1\tag{5}$$

The roots of the quadratic equation are

$$z_{1,2}=-\frac{a_1}{2}\pm\sqrt{\frac{a_1^2}{4}-a_2}\tag{1}$$

For $a_1^2/4\ge a_2$, the roots are real-valued. In that case we require

$$-1<z_{1,2}<1\tag{2}$$

Let's start with the first inequality (from the left) in $(2)$:

$$-1+\frac{a_1}{2}<-\sqrt{\frac{a_1^2}{4}-a_2}\tag{3}$$

Note that the minus sign before the square root represents the more restrictive constraint.

Squaring $(3)$ (note that $<$ becomes $>$) and rearranging gives

$$a_1-a_2<1\tag{4}$$

In the same way you can obtain the other inequality by considering the second inequality in $(3)$ (with a positive sign before the square root):

$$a_1+a_2>-1\tag{5}$$

The roots of the quadratic equation are

$$z_{1,2}=-\frac{a_1}{2}\pm\sqrt{\frac{a_1^2}{4}-a_2}\tag{1}$$

For $a_1^2/4\ge a_2$, the roots are real-valued. In that case we require

$$-1<z_{1,2}<1\tag{2}$$

Let's start with the first inequality (from the left) in $(2)$:

$$-1+\frac{a_1}{2}<\pm\sqrt{\frac{a_1^2}{4}-a_2}\tag{3}$$

Squaring $(3)$ and rearranging gives

$$a_1-a_2>1\tag{4}$$

In the same way you can obtain the other inequality by considering the second inequality in $(3)$:

$$a_1+a_2>-1\tag{5}$$

The roots of the quadratic equation are

$$z_{1,2}=-\frac{a_1}{2}\pm\sqrt{\frac{a_1^2}{4}-a_2}\tag{1}$$

For $a_1^2/4\ge a_2$, the roots are real-valued. In that case we require

$$-1<z_{1,2}<1\tag{2}$$

Let's start with the first inequality (from the left) in $(2)$:

$$-1+\frac{a_1}{2}<\pm\sqrt{\frac{a_1^2}{4}-a_2}\tag{3}$$

Squaring $(3)$ and rearranging gives

$$a_1-a_2<1\tag{4}$$

In the same way you can obtain the other inequality by considering the second inequality in $(3)$:

$$a_1+a_2>-1\tag{5}$$