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QUESTION: Here are the equations:

x^2+y=62

x+y^2=176

I have tried to substitute the value of y from one equation in another. This leads to a quartic equation. Eliminating co-efficients seems to be difficult.

ANSWER: Is this from a book or something? If you sketch the two parabolas carefully you will see they intersect in 4 places, so it is indeed a quartic equation, which doesn't in general have any nice solutions. Do you need exact answers? You can certainly estimate them however accurately you like.

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QUESTION: I was curious to understand whether this equation can be solved algebraically

without too much complexity. How to obtain real values of roots?

There's a formula for solving cubic and quartic equations but it is very messy. Occasionally you get lucky and a quartic factors nicely, but that is not the usual situation.

For degrees higher than 5 there is no such formula, this is an important theorem from Galois theory.

The formula is horrible, you can see it here:

http://planetmath.org/QuarticFormula.html

Of course a good graphing calculator or computer can quickly estimate the roots, for example using Newton's method.

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