Solve a^2+2ageq0

$${ a }^{ 2 } +2a \geq 0$$

Solve for a

$$a\in (-\infty,-2]\cup [0,\infty)$$

Solution Steps

Factor out a.

$$a\left(a+2\right)\geq 0$$

For the product to be ≥0, a+2 and a have to be both ≤0 or both ≥0. Consider the case when a+2 and a are both ≤0.

$$a+2\leq 0 a\leq 0$$

The solution satisfying both inequalities is a\leq -2.

$$a\leq -2$$

Consider the case when a+2 and a are both ≥0.

$$a\geq 0 a+2\geq 0$$

The solution satisfying both inequalities is a\geq 0.

$$a\geq 0$$

The final solution is the union of the obtained solutions.

$$a\leq -2\text{; }a\geq 0$$