A Nice Math Problem | How to solve for X in this Olympiad Problem ?

[@ManjulaMathew-wb3zn]

I think it’s a great equation to solve. The question should have stated x is a real number. So there are no domain restrictions for X. However m should always be positive coz it’s the square of x. In this case both 2 and the positive radical value providing two positive and two negative roots. The highest power of x is 8. Since x is real 4 roots are correct.

[@ccudmore]

You reject the second option because you state that x must be a positive integer but then accept the first solution of root 2 - which is not an integer

[@muqaddasadnan1743]

If x is a positive integer then how sqrt 2 is an integer?

[@BP-gn2cl]

This is a very long, tedious and not logically concluded process.

[@ManjulaMathew-wb3zn]

What I really value is the method. The substitution of n divides the expression in to 2 factors each having 4th power of the variable. Then the substitution of m split each of the above factors in to two quadratic factors. Good approach overall.

[@BP-gn2cl]

Reject t=-3, (1-√29) /2 outrightly Then reject t=(1+√29) /2 as it will give x^4>7

[@에스피-h8t]

Solution by insight Let x^2=t t^2+rt(t+7)=7 2^2+rt9=7 t^2=2 x=rt2 or -rt2

[@muqaddasadnan1743]

Is √2 is an integer?

[@samuelbenet007]

Peux-tu expliquer pourquoi x devrait être un nombre entier ?

[@BP-gn2cl]

Just substitute x^2=t

[@BP-gn2cl]

Also x^2=(1+√29) /2 value is rejected because in that case x^4>7

[@cyberxnova]

DIFFERENT APPROACH ( find the mistake challenge ) x^4 + √(x^2 + 7) = 7 x^2 = [7 - √(x^2 + 7)]^2 x^2 = 49 + x^2 + 7 -14√(x^2 + 7) x squrt will cancel out and after furthur solving we will get, √(x^2 + 7) = 56/14 = 4 x^2 = 16 -7 = 9 x = √9 x = ±3 Challenge -- find mistake

[@samuelbenet007]

Il faut dire x = ±√9 (et non x = √9 , car √9 = 3 (jamais -3) !! Sinon (±3)^4 + √((±3)^2 + 7) = 85 (et non 7) ^^

[@BP-gn2cl]

Your mistake is in the first step X^2=[7-(√x^2+7)]^1/2 And instead of power 1/2 u have written 2.

[@muqaddasadnan1743]

If x>0 then how we will take -√2

[@newtime1811]

m >0 , not x >0

[@BP-gn2cl]

U will get t=2, -3, (1+/-√29) /2

[@BP-gn2cl]

Very long and tedious process.

[@BP-gn2cl]

So only real solutiins are x=+/-√2

[@BETEP_BECT]

Х^Х=7 ?

[@samueldeandrade8535]

That's wrong.

[@samuelbenet007]

Le résultat semble être bon, mais dans son raisonnement, il y a une (ou plusieurs) erreur(s)

[@АндрейКожевников-о8й]

Х= Корень из 2.

[@Misha-g3b]

x=+_V2.