non linear system of equations leading to quartic

Question

The question asks to solve the system

$$\begin{cases} \sqrt x + y = 16 \\ \sqrt y + x = 25 \end{cases}$$

Substitution leads to a fourth degree polynomial.

Is there n easier way to solve it ?

Thanks.

Answer 1

There is no simpler way.

If there was one, you would have found a new and simpler method to solve quartic equations, and this is not true.

By the way, the exact expressions as given by a CAS are terrible, and no simple method leads you to a terrible expression.

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There is a more geometrical way to address this question. Both equations represent parabolas (replacing the unknowns by their squares). It is well-known that to intersect two conics you can consider the pencil they form (linear combination of the equations), and by a suitable choice of the coefficients you make the combination degenerate in two straight lines. Then the problem is reduced to twice the intersection between a conic and a straight line (quadratic problems).

Unfortunately, for the same reason as above, finding the degenerate conic leads to a cubic equation (as there are three solutions, corresponding to all ways to take the intersection points in pairs). In fact, if you are familiar with the resolution of quartic equations, you will recognize the reduction to a cubic followed by the resolution of two quadratics. So no shortcut.

Answer 2

The quartic is as follows:

$x^{4}-100x^{3}+3718x^{2}-60901x+370881=0$

The solutions are:

$x_{1}=-a-b+c+25$,

$x_{2}=-a+b-c+25$,

$x_{3}=+a-b-c+25$,

$x_{4}=+a+b+c+25$;

where

$a=\frac{\sqrt{3}\sqrt{ 16-\sqrt{949} .cos\Big( \frac{atan\big(f\big)}{3}+\frac{\pi}{3} \Big)}} {3}$;

$b=\frac{\sqrt{3}\sqrt{ 16-\sqrt{949}.sin\Big( \frac{acot\big(-f\big)}{3}\Big)}} {3}$;

$c=\frac{\sqrt{3}\sqrt{ 16+\sqrt{949}.cos\Big( \frac{atan\big(f\big)}{3}\Big)}} {3}$;

$f=\frac{3\sqrt{108079791}}{466715}$