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Using calculus to generate the quadratic formula

By TwoPi

I get (unreasonable considerable) joy in finding fancy new proofs of elementary results, proofs that might come under the heading of mathematics made difficult or obscure. (I’ve never seen Linderholm’s book, but suspect it would appeal to a twisted part of my psyche.) I don’t quite understand the psychology behind this liking, but there it is nonetheless.

One of my favorite examples follows: solving quadratic equations using integration to complete the square.

Suppose f(x)=ax^2+bx+c, and we want to find the solutions to f(x)=0.

Note that f'(x) = 2ax+b and f(0) = c, and thus it must be that f(x) = c+ \int_0^x 2at+b \; dt .

We compute this antiderivative using the change of variables w=2at+b, \quad dw = 2a\;dt, which leads to c + \int_b^{2x+b} \frac1{2a} \; w \; dw. This last expression is equal to c + \left( \frac1{4a} (2ax+b)^2 - \frac{b^2}{4a} \right).

Thus the roots of f(x) = 0 are found by solving the equation c + \frac{ (2ax+b)^2 - b^2 }{4a} = 0.

This leads to (2ax+b)^2 -b^2 = -4ac, and thus 2ax+b = \pm \sqrt{b^2 - 4ac}, and so finally we arrive at the roots x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{2a} .

Forthcoming: Using Fubini’s Theorem to prove Integration by Parts

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This entry was posted on October 19, 2008 at 7:12 pm and is filed under Miscellaneous, Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

4 Responses to “Using calculus to generate the quadratic formula”

  1. Integration by parts via iterated integrals « 360 Says:
    October 22, 2008 at 4:57 pm | Reply

    [...] of an occasional series, featuring obfuscated proofs of elementary results.  Previously: Using calculus to generate the quadratic formula.  [...]

  2. 42nd Carnival of Mathematics — The Endeavour Says:
    October 24, 2008 at 5:32 am | Reply

    [...] at the 360 blog presents two fun articles using advanced techniques to prove elementary results, Using calculus to derive the quadratic equation and Integration by parts via iterated [...]

  3. trevorpythag Says:
    February 2, 2009 at 1:42 pm | Reply

    much more interesting than the “completing the sqaure” proof we learnt in school!!

  4. AP Calculus AB: Tuesday, 2/17/2009 « Dr. Cooper’s Wyoming Seminary Math Weblog Says:
    February 17, 2009 at 1:50 pm | Reply

    [...] Challenge problem: Derive the quadratic formula using integration. The solution is here. [...]

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