A frequentist does his maths homework

Question 1: Solve the quadratic equation x^2 + 4x - 1 = 0.

Answer: The two solutions are given by the quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a=1, b=4, and c=-1. Therefore, after some simplification, the two solutions are x=-2 + \sqrt{5} \approx 0.236 and x=-2 - \sqrt{5} \approx -4.236.

I don’t like much how there is an ambiguity here. It seems to suggest we’d need more information if we actually wanted to know x in a real problem. Sure, if we knew x was positive, we’d take the positive solution, and conversely if it’s negative. But how would we ever know that?

To solve this problem, I propose an alternative methodology. Let’s assume x=-2 + \sqrt{5} \approx 0.236 is actually the solution. From this we can prove that x^2 + 4x - 1 = 0And since zero is greater than -1, we can also say x^2 + 4x - 1 > -1Yep. I propose this as a general procedure for solving quadratic equations. Using the quadratic formula, take the root with the plus sign, not the minus sign. Then use it to derive an inequality that you know for certain is true. This way, you don’t need to
rely on extra assumptions to determine which root is correct, and there is no ambiguity, unlike in the \pm religion.

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About Brendon J. Brewer

I am a senior lecturer in the Department of Statistics at The University of Auckland. Any opinions expressed here are mine and are not endorsed by my employer.
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