Mathematical Elegance Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Compare: (A) solving x2−5x+6=0x^2-5x+6=0 by the quadratic formula, (B) factoring as (x−2)(x−3)=0(x-2)(x-3)=0. Which is more elegant?

Solution

  1. 1
    Method A (quadratic formula): x=5±25−242=5±12x = \frac{5 \pm \sqrt{25-24}}{2} = \frac{5 \pm 1}{2}, giving x=3x=3 or x=2x=2. Correct but computational.
  2. 2
    Method B (factoring): spot that (−2)(−3)=6(-2)(-3)=6 and (−2)+(−3)=−5(-2)+(-3)=-5. Immediately: (x−2)(x−3)=0(x-2)(x-3)=0, so x=2x=2 or x=3x=3. One line.
  3. 3
    Elegance: Method B is more elegant — it uses insight rather than formula, and is faster and more transparent.

Answer

x=2 or x=3; factoring is more elegantx=2 \text{ or } x=3;\text{ factoring is more elegant}
When factoring works, it is generally more elegant than the quadratic formula because it is shorter, more insightful, and directly reveals the roots' structure. The formula is more powerful (works always) but less elegant.

About Mathematical Elegance

The aesthetic quality of a mathematical argument or result that achieves its goal with striking simplicity, insight, or economy of means.

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