Which of the following equations has 1 + √2 as one of its roots? GMAT Problem Solving

\(​​\)Question: Which of the following equations has \(1+\sqrt{2}\) 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. \(x^2-x-1=0\)

Answer:
Approach Solution (1):
To solve this problem, we need to use the following two facts:
(1) If a quadratic equation has integers coefficients only, and if one of the roots is \(a+\sqrt{b}\) (where a and b are integers), then \(a-\sqrt{b}\) 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 \(1-\sqrt{2}\) is a root of the quadratic equation, we can let:
r =\(1+\sqrt{2}\)
and
s =\(1-\sqrt{2}\)
Thus, r + s =\((​​1+\sqrt{2})+(1-\sqrt{2}) = 2\) and \(rs = (1+\sqrt{2})(1-\sqrt{2})=1-2=-1\)
The quadratic equation must be \(x^2-2x-1=0\)

Correct option: D

Approach Solution (2):
\(x=1+\sqrt{2}\)
\(x-1=\sqrt{2}\)
Squaring both sides:
\((x-1)^2=(\sqrt2)^2\)
\(x^2+1-2x=2\)
\(x^2-2x-1=0\)

Correct option: D

Approach Solution (3):
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\), using equation for finding roots of the quadratic equation
Where b is the coefficient of x, a is the coefficient of \(x^2\) and c is constant
Substituting values for option A gives
\(x = -1 \pm {\sqrt{2}}\)
Since we need root \(1+\sqrt{2}\) as, b must be negative, with other coefficient same as A which is option D

Correct option: D

“Which of the following equations has \(1+\sqrt{2}\) as one of its roots?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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