Convert a Quadratic Formula to Roots Form by Completing the Square

There are many methods to do this but there are few shorter than this one.


The Tutorial

  1. Write down each formula step please. Let's start with ax^2 +bx +c = y = 0. We are setting the y value to 0 to find the intercepts on the x axis. where y=0.
  2. Subtract c from both sides and obtain ax^2 + bx = -c
  3. Multiply both sides by 4a to obtain 4a^2x^2 + 4abx = -4ac
  4. Complete the square on the left and add b^2 to the right side: (2ax + b)^2 = b^2 - 4ac. You may want to multiply (2ax + b)^2 out to make sure everything is OK. It's a good practice to follow.
  5. Take the square root of both sides to obtain (2ax + b) = ± sqrt(b^2 - 4ac)
  6. Subtract b from both sides, then divide both by 2a to obtain x = (-b ± sqrt(4ac))/2a.

Explanatory Charts, Diagrams, Photos

  1. Since there are two x roots, depending on the ±, restate as {x1, x2} = (-b ± sqrt(4ac))/2a.
  2. If you find a shorter version, please let me know.
  3. It's good practice to also work it backwards to the original form, i.e. derive the Quadratic Formula. Try that now if you please.

n.b. The Quadratic Relation of Neutral Operations / Symmetry by Commutation

  1. Open an Excel worksheet and take notes; save the file as something like Quadratic NeuOps into a logical folder.
  2. Note that the Neutral Operation x+b = x*b = c holds the two operators, Addition and Multiplication, neutral with regard to the set {x, b, c | x or b ≠ 1} (else division by 0 results in the denominator, which is either undefined or infinite, where Infinity is not a number.)
  3. Work through the easy few steps again, realizing that both Addition and Multiplication are commutative, so what applies for x, applies equally to b -- there is Symmetry.
      • x+b-b = xb-b
      • x = b*(x-1)
      • x/(x-1) = b and b/(b-1) = x, by commutation. We have isolated and defined b in terms of x and 1, where x may not equal 1, and b may not equal 1. Given x then, we may determine b.
      • Let us now substitute x/(x-1) for b in the original equation:
      • x+ x/(x-1) = x * x/(x-1), and the right becomes x^2/(x-1) = c, or b^2/(b-1) = c.
      • Distribute the left's denominator to c by multiplying both sides by it, to get:
      • x^2 = cx - c, or stated in ax^2 + bx + c = y = 0 form, you get 1x^2 - cx + c = 0. a=1, b=c.
      • Stated in roots form, you should arrive at:
      • {x1, x2} = ( -(-c) ± sqrt(c^2 - 4*1*c))/(2*1)
      • Let c = 1 and the result is imaginary. More interesting to this editor is c = 5, like the 5 fingers on your hand -- exactly like the 5 fingers on your hand let's say.
      • {x1} = (5 + sqrt(25 - 20))/2 = 3.618033989, and Phi is clearly visible!
      • {x2} = (5 - sqrt(25 - 20))/2 = 1.381966011, which is Phi^2 +1 !!
      • Each finger on your hand is proportioned according to Phi, please realize. Not by what was just proved here, but it is a known scientific fact. That is some Sacred Geometry, note well.
      • Lastly, what was true on the one hand for x, is equally true on the other hand for b, due to Symmetry by Commutation.
    • n.b. It is one's belief that, etymologically, the word "five" and "Phi" are related at root, probably at the morphemic level, "phi" being of Greek meaning for you to research on your own at this point. Excitedly, one hopes!

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the Related Articles below and the article How to Do the Sub Steps of Neutral Operations for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation relating to Neutral Operations.
    • For more art charts and graphs, you might also want to click on Algebra, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.


  • =sqrt(5) is simply Excel's way of expressing the square root of 5 in formula form.

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