MHB How Do You Solve This Complex Quadratic Equation?

AI Thread Summary
To solve the complex quadratic equation $\frac{x^2+2}{x}+\frac{8x}{x^2+2}=6$, the discussion suggests simplifying it to $x^4-6x^3+12x^2+12x+4=0$. By substituting $y = \frac{x^2+2}{x}$, the equation transforms into $y + \frac{8}{y} = 6$, which can be solved as a quadratic in y. This approach allows for finding y first, and subsequently determining the values of x. The overall consensus is that the problem can be tackled without exceeding quadratic complexity.
paulmdrdo1
Messages
382
Reaction score
0
please help me with this

$\frac{x^2+2}{x}+\frac{8x}{x^2+2}=6$

this is where I can get to when I simplify the the equation above,

$x^4-6x^3+12x^2+12x+4=0$
 
Mathematics news on Phys.org
paulmdrdo said:
please help me with this

$\frac{x^2+2}{x}+\frac{8x}{x^2+2}=6$

this is where I can get to when I simplify the the equation above,

$x^4-6x^3+12x^2+12x+4=0$

the above has become more complex
in case you put
$\frac{x^2+2}{x}= y$

then you get
$ y +\frac{8}{y} = 6$

you get quadratic in y then solve for y and based on it solve for x

I hope you can proceed because at no stage you get more than quadratic
 
paulmdrdo said:
please help me with this

$\frac{x^2+2}{x}+\frac{8x}{x^2+2}=6$

this is where I can get to when I simplify the the equation above,

$x^4-6x^3+12x^2+12x+4=0$

Let $ y = \frac{x^2+2}{x} $
Solve it for y
$y + \frac{8}{y} = 6 $
Then solve it for x
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Back
Top