A Rational Equation | Can you solve?
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- Опубліковано 11 лип 2024
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2=x was mental arithmetic by substitution.
Other method: split 8 as 4+4, make 2 differences of squares: x^2 - 4 + (x/(x-1))^2 - 4. After decomposing, (x-2) is common factor and you go straight to 3rd degree equation, where x=2 is again a rational solution.
Nice one...
Thanks 😊
Same as already 2-3 times before!
I whould have started with domain analyse x^2 + [ x/(x-1)]^2 = 8 lhs is not x = 1 and x^2 + 1 is the limit as x -> +/- oo check x^2 = 4 this gives one solution at x=2.
The multiple out to get the quartic then factor via division {x-2} this reduces to a quadratic which is then directly solvable.
Syber, I know you like to use minus-plus instead of plus-minus, and I'm fine with that. Would you then use plus-minus in the sum and difference cosine formulas to show to indicate that the signs should be opposite? In the textbooks I'm familiar with, minus-plus is encountered when corresponding opposite signs are needed.
Did I use them with trig sums?
There are instances (like R.R.T) where the plus/minus is unordered, meaning for two plus/minus's in a formula (say from using the quadratic formula twice) you would have 4 different combinations to consider. There are also instances, like the trig sum formulas, where they are ordered, meaning you have to only take the top symbols or only take the bottom symbols to have a true equation. In whatever context, the meaning of plus-minus will have to be stated as part of the theorem.
Though in the trig sum case, the use of minus plus for that purpose is more of a notational convenience for equations that only differ in that one case, I'd guess? In practice you're not interested in solving for both the sum and the difference simultaneously, and it would be easy to lose track of the sum and difference as you manipulate the equation. Not sure if there's any other instances where "ordered" minus/plus is actually used, so you can probably just use them interchangeably and the context would be implied to be that the plus/minus order doesn't matter and every combination is considered, unless stated otherwise.
@@looney1023 Exactly. It cuts the number of formulas to memorize by half. If I actually said it would be impossible to memorize the formulas individually, I'd be lying; there are cases where sign differences must be memorized individually. Consider the derivatives and integrals of the trig functions and the derivatives of the hyperbolic functions.
@@SyberMathcan you do some related rates and optimization problems
Sybermath tries to avoid the issue you are raising concerning his notation by denying that he uses ∓ with trig identities, but that is beside the point. There are other identities which use both ± and ∓ to indicate opposite signs, e.g.
a³ ± b³ = (a ± b)(a² ∓ ab + b²)
(a² + b²)(c² + d²) = (ac ± bd)² + (ad ∓ bc)²
👍
Substitution = 42 = meaning of life
x = 2
X = 2, -1 +- sqrt(3)
Yes.
nice! 😁
x=2, -1±√3