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

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To solve the equation 600*sqrt{1-a^2/400}=578-a^2/2, one effective method is to substitute a^2 with x and square both sides, resulting in quadratic polynomials. Simplifying the left-hand side to 30*sqrt{400-a^2} can streamline the process. Multiplying both sides by 2 helps eliminate fractions, making calculations easier. It's crucial to check that x remains less than or equal to 400 to avoid extraneous roots from squaring. Overall, these strategies can lead to a more elegant solution.
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600*sqrt{1-a^2/400}=578-a^2/2

Shorter and elegant tricks would be welcome!
 
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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 $y$.
 
Evgeny.Makarov said:
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 :)
 

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