I'm supposed to prove that all the solutions to X^2 = X + 1 are irrational.
I'm not sure exactly where to begin. I know by contradiction I would make it rational and replace x with A/B. Then work the problem through. After I worked it through I ended up with
0 = b/a + (b^2)/(a^2) and I'm not even sure if i'm on the right track even.
Do you any of you know?
Thanks for the help.
Well heres what I've done. Hoepfully it sounds about right. I actually get this roots test better then some of the area material. If this sounds wrong please tell me.
Proof that all the solutions to the equation are irrational. (hint – rational/irrational numbers)
Theorem: All solutions to the equation are irrational.
Proof: To solve this equation, let the equation = 0, so that the new equation is –X^2 + x + 1.
Then use the Rational Roots test to find any of the rational numbers that will make the equation = 0.
By using the Rational Roots test we find the factors of the first and last part of the equation. By doing this we get -1, 1.
By using the two rational numbers found in the equation we find that none equal to 0. Meaning that there are no rational solutions to the equation, thus there is an irrational solution to the equation and the conclusion is true.