Which of the following equations has \(1 + \sqrt2\) as one of its roots?

[Gmat_mission]

Which of the following equations has \(1 + \sqrt2\) as one of its roots?

A) \(x^2 + 2x – 1 = 0\)
B) \(x^2 – 2x + 1 = 0\)
C) \(x^2 + 2x + 1 = 0\)
D) \(x^2 – 2x – 1 = 0\)
E) \(x^2 – x – 1= 0\)

[spoiler]OA=D[/spoiler]

Source: Official Guide

[Scott@TargetTestPrep]

Which of the following equations has \(1 + \sqrt2\) as one of its roots?

A) \(x^2 + 2x – 1 = 0\)
B) \(x^2 – 2x + 1 = 0\)
C) \(x^2 + 2x + 1 = 0\)
D) \(x^2 – 2x – 1 = 0\)
E) \(x^2 – x – 1= 0\)

[spoiler]OA=D[/spoiler]

Source: Official Guide

Solution:

To solve this problem, we need to use the following two facts:

1) If a quadratic equation has integer coefficients only, and if one of the roots is a + √b (where a and b are integers), then a - √b is also a root of the equation.

2) If r and s are roots of a quadratic equation, then the equation is of the form x^2 – (r +s)x + rs = 0.

Since we know that 1 - √2 is a root of the quadratic equation, we can let:

r = 1 + √2

and

s = 1 - √2

Thus, r + s = (1 + √2) + (1 - √2) = 2 and rs = (1 + √2)(1 - √2) = 1 – 2 = -1.

Therefore, the quadratic equation must be x^2 – 2x – 1 = 0.

Answer: D