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I am a math tutor looking for a better way to explain quadratics to my students.

I came across the so-called "Po-shen Loh" method on YouTube, and I tried to apply it to a quadratic equation that one of my students was having trouble with. (Here is the video that I learned it from, skipped to the example that is most similar to my problem: https://youtu.be/XKBX0r3J-9Y?t=1663)

The equation that I want to solve is $5x^2 - 4x - 6 = 0$

Here is the correct result, as confirmed by symbolab (https://www.symbolab.com/solver/quadratic-equation-calculator/solve%20for%20x%2C%205x%5E%7B2%20%7D-4x-6%3D0):

$$ x_{1,2} = \frac{2 \pm \sqrt{34}}{5} $$

However, when I try to work it out using the "Po-shen Loh" method, I keep getting this result:

$$ x_{1,2} = \frac{2 \pm \sqrt{40}}{5} $$

Here are my calculation steps, please help me if you can: My calculation, apparently incorrect

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    $\begingroup$ $(2/5 - u)(2/5 + u) \ne 2/5 - u^2$. $\endgroup$
    – player3236
    Mar 4, 2021 at 10:27
  • $\begingroup$ Thank you! That worked! You found my mistake so quickly! $\endgroup$
    – TookieWookie
    Mar 4, 2021 at 10:31
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    $\begingroup$ For what it is worth. I totally disagree with teaching students this. Unlike equations of degree higher than 2, quadratic equations can be totally conquered by explaining both the analysis and intuition behind $$Ax^2 + Bx + C = 0 \implies x = \frac{1}{2A}\left[-B \pm \sqrt{B^2 - 4AC}\right].$$ I would leave the methods in the youtube to be methods that students work out on their own after they have applied the quadratic equation above in approximately 100 different problems. Reason: the above formula is simpler (in general) than the theory behind the methods in the youtube video. $\endgroup$
    – user2661923
    Mar 4, 2021 at 10:42
  • $\begingroup$ I was looking for a method that was more intuitive to apply, to make it easier for people who seriously struggle with math. @user2661923, it seems that the normal quadratic formula that is taught (which you quote) is indeed easier after all. But still, people take one look at all that algebra and they freak out... $\endgroup$
    – TookieWookie
    Mar 4, 2021 at 10:46
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    $\begingroup$ Probably because they aren't taught the quadratic formula correctly. That would entail, starting with the motivation behind completing the square, then understanding (without memorizing) the idea(s) behind completing the square, then understanding (without memorizing) the quadratic formula itself. Then, only after the student has a deep understanding of the quadratic formula, would I have the students then memorize it. Then, I would give examples where [Ax^2 + Bx + C] either intersects the $x$-axis in exactly 1 point, or in exactly 0 points, and discuss the roots of these equations. $\endgroup$
    – user2661923
    Mar 4, 2021 at 10:51

1 Answer 1

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(I am eternally grateful to player3236)

My mistake was in the line:

$\frac{2}{5} - u^2=-\frac{6}{5}$

However, this should have been:

$\frac{4}{25} - u^2=-\frac{6}{5}$

Why? In the immortal words of player3236:

$(2/5 - u)(2/5 + u) \neq 2/5 - u^2$

The "Poh-shen Loh" method involves a difference of squares at that stage in the calculation.

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