I already know how to solve this after the answer being told to me, just wondering how others would even approach the problem from their perspective. Like imagine you got to solve this equation at an olympiad, where would you start?
#Already solved, just training with problem solving, polynomial equation
crimson quarry
slender barnBOT
crimson quarry
that would be lovely
my first instinct was the most simple just square both sides but then you end up with a 4th degree polynomial
which sucks
tardy yacht
well that is true but let me solve it first so it will be easier for me to explain
crimson quarry
kay
tardy yacht
So far ive been able to move everything to one side and then use rrt to just work it
lime widget
I would probably square both sides to get rid of the square root, but it ends up being a quartic equation, which are not easy to solve
tardy yacht
Rational root theorem
and then just try one or two times
Thats the only non precalc way I know how to solve it
Other wise you can use polar form to simplify this equation but that would not be algebra only
crimson quarry
the solution given by the olympiad is damn crazy
it's in lithuanian though so I'll have to translate it
tardy yacht
and if you find it hard to expand theres a really easy way to setup polynomial expansion such that you just have to add the diagonals
It also works for any form of polynomial equation
And for any multiplication
so
??
crimson quarry
patience
alright don't know how the formatting will go for this but here
- Solve equation $2x^2 + 6x + 9 = 7x\sqrt{2x + 3}$
Solution. Because for all x the inequality is true $2x^2 + 6x + 9 > 0$, that means $x > 0$. Marking $u = \sqrt{2x + 3}$, the equation becomes $2x^2 + 3u^2 = 7xu$. From this $(2x^2 - ux) + (3u^2 - 6ux) = 0, x(2x - u) + 3u(u - 2x) = 0, (2x - u)(x - 3u) = 0$. That means $2x - u = 0$ or $x - 3u = 0$. In the first case we get the equation $2x = \sqrt{2x + 3}, 4x^2 = 2x + 3$, the positive solution for the quadratic equation is $x_1 = \frac{1+\sqrt{13}}{4}$, in the second case $x = 3\sqrt{2x + 3}, x^2 = 18x + 27$, the positive solution for this equation is $x_2 = 9 + 6\sqrt{3}$
Solution: [the first and second solution]
brittle bloomBOT
A guy named Karolis
