MHB How to solve this dreadful quadratic with paper and pen?

[Sawarnik]

600*sqrt{1-a^2/400}=578-a^2/2

Shorter and elegant tricks would be welcome!

 

[Evgeny.Makarov]

The straigtforward way is to denote $a^2$ by $x$ and take the square of both sides. Then both sides become quadratic polynomials. After finding $x$, it is necessary to check that $x\le 400$ to avoid gaining extra roots during squaring. The left-hand side can be simplified to $30\sqrt{400-a^2}$; then both sides can be multiplied by 2 to avoid fractions. It may help a little to denote $400-x$ by $y$.

 

[Sawarnik]

The straigtforward way is to denote $a^2$ by $x$ and take the square of both sides. Then both sides become quadratic polynomials. After finding $x$, it is necessary to check that $x\le 400$ to avoid gaining extra roots during squaring. The left-hand side can be simplified to $30\sqrt{400-a^2}$; then both sides can be multiplied by 2 to avoid fractions. It may help a little to denote $400-x$ by $y$.

Thanks :)