Solving Quadratic Equations with Radicals
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$$x^2 - 2x - 1 = 0$$
The equation above has solutions $x = n + \sqrt{k}$ and $x = n - \sqrt{k}$, where $n$ and $k$ are positive integers.
What is the value of $n + k$ ?
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Step by Step Written Solution
Let's find the values of n and k for this quadratic equation and then calculate their sum.
Solving $x^2 - 2x - 1 = 0$
The equation is given as x squared minus two x minus one equals zero. We can use the quadratic formula to solve it.
First, let's identify our coefficients: a is one, b is negative two, and c is negative one.
Now, we plug these into the quadratic formula: x equals negative b, plus or minus the square root of b squared minus four a c, all over two a.
Substituting our values, we get x equals negative negative two, which is positive two, plus or minus the square root of negative two squared minus four times one times negative one, all divided by two.
Substitution & Simplification
Simplifying the terms, negative two squared is four, and negative four times one times negative one is positive four.
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