Solving 4th Degree Equation: √x+y=7, √y+x=11

[obing]

√x+y=7
√y+x=11
solving this equation using x=-b√b2-4ac/2a for finding the root is impossible

the equation turns out to be of 4th degree

or two quadratic eq. can anybody help

this isn't a homework , just wanted to know ...;)

 

[Mentallic]

If you're trying to solve for x in these equations, isolate y in either the first or second and then substitute it into the other equation.

$\sqrt{x}+y=7$ (1)

$\sqrt{y}+x=11$ (2)

Rearranging (2) to make y the subject:

$\sqrt{y}=11-x$

$y=(11-x)^2$

Now substitute into (1):

$\sqrt{x}+(11-x)^2=7$

To simplify this it's important to try get rid of the square root. This would imply squaring, but remember if we just squared both sides as it is, something like $(\sqrt{a}+b)^2=a+2b\sqrt{a}+b^2$ so we don't really get rid of the root, we just create more problems.

So instead, isolate the $\sqrt{x}$ and then square both sides.

Yes you will get a quartic (4th degree) equation and it is generally very complicated to solve, unless you're in luck and this question was devised to have nice, rational roots in which case you could then use the rational root theorem to find them.
But I doubt this since it sounds like you created the problem yourself.

 

[Rasalhague]

http://www.wolframalpha.com/input/?i=sqrt%28x%29%3D7-%2811-x%29^2

According to this, one real solution, 9, and one complex solution, which I won't attempt to copy...

 

[Mentallic]

Certainly, x=9 is a solution.

If you continue to solve and finally end with the quartic and then solve that for x, you're given 4 real solutions. Some of these however could be extraneous solutions.

But look at what wolfram gives as solutions to the original system of equations:
http://www.wolframalpha.com/input/?i=sqrt(x)+y=7,+sqrt(y)+x=11"

Also, graphmatica draws $\sqrt{y}+x=11$ as the parabola $y=(11-x)^2$ which is fine except that for x>11 it does not satisfy the first equation. And this really stumped me at first.

Well, I'm stumped. Is the only solution x=9 or are there more complex solutions?

 

[obing]

thanks guys :)

ive found a way out


if we take x= l2

and y=n2

then √l2 +y= 7 (√x+y=7)


√n2+x =11 (√y+x=11)

which becomes l2+y-7=0 (getting rid of root)



n2+x-11=0

2 quadratic equn. :)

ans sloving it gives x=9 y=4