Find Quadratic Equation Given 1 Root: (2^(1/2)+1)

[ritwik06]

Homework Statement

If one root of a quadratic equation with rational co-efficient is (2^(1/2)+1) , then find the quadratic equation.

Homework Equations

x=(-b+d^1/2)/2a
d=b^2-4ac

The Attempt at a Solution

Well, I can't quite understand this question. Please help me to understand what is given in the following statemnt.

 

[HallsofIvy]

If the only square root in the solution is $\sqrt{2}$, what must d be? Since the only difference between roots of a quadratic equation is that $\pm$ before the square root, what must the other root be?

Another, more "sophisticated" method:
Any quadratic, $ax^2+ bx+ c$ can be written $a(x- x_0)(x- x_1)= ax^2- a(x_0+x_1)+ ax_0x_1$ where $x_0$ and $x_1$ are roots of the equation. If one root is $1+ \sqrt{2}$ what must the other be so that both $x_0+ x_1$ and $x_0x_1$ are rational?

 

[ritwik06]

Well, Please check this. I think the othr root is 1-(2^1/2). thre fore the quadratic equation is x^2-2x-1=0
Am I right?

 

[HallsofIvy]

Yes, that is correct.