Solve |2020-a|+sqrt{a-2a2}=a

$$| 2020 - a | + \sqrt { a - 2 a 2 } = a$$

Solve for a_2

$$\left\{\begin{matrix}a_{2}=-2a^{2}+\frac{8081a}{2}-2040200\text{, }&2a-2020\geq 0\text{ and }a\leq 2020\\a_{2}=\frac{a}{2}-2040200\text{, }&a\geq 2020\end{matrix}\right.$$

Steps for Solving Radical Equations

Subtract |2020-a| from both sides of the equation.

$$\sqrt{-2a_{2}+a}+|2020-a|-|2020-a|=a-|2020-a|$$

Subtracting |2020-a| from itself leaves 0.

$$\sqrt{-2a_{2}+a}=a-|2020-a|$$

Subtract |2020-a| from a.

$$\sqrt{-2a_{2}+a}=-|2020-a|+a$$

Square both sides of the equation.

$$-2a_{2}+a=\left(-|2020-a|+a\right)^{2}$$

Subtract a from both sides of the equation.

$$-2a_{2}+a-a=\left(-|2020-a|+a\right)^{2}-a$$

Subtracting a from itself leaves 0.

$$-2a_{2}=\left(-|2020-a|+a\right)^{2}-a$$

Divide both sides by -2.

$$\frac{-2a_{2}}{-2}=\frac{\left(-|2020-a|+a\right)^{2}-a}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$a_{2}=\frac{\left(-|2020-a|+a\right)^{2}-a}{-2}$$

Divide \left(a-|2020-a|\right)^{2}-a by -2.

$$a_{2}=\frac{-\left(-|2020-a|+a\right)^{2}+a}{2}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{16161-32a_{2}}+8081}{8}\text{, }&a_{2}\geq -2039190\text{ and }a_{2}\leq \frac{16161}{32}\\a=\frac{-\sqrt{16161-32a_{2}}+8081}{8}\text{, }&a_{2}\geq 505\text{ and }a_{2}\leq \frac{16161}{32}\\a=2a_{2}+4080400\text{, }&a_{2}\geq -2039190\end{matrix}\right.$$