Jeff Miller

GMAT OFFICIAL GUIDE PS – Which of the following equations has 1 + √2 as one of…

Which of the following equations has 1 + √2 as one of…

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

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