Vertex Form Quadratic Written Using Factored Form Variables

Do you prefer this method or the traditional method for converting quadratics?

  • This Method

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Oct 2014
Ottawa, ONT, Canada
Typically, the conversion of a factored form quadratic into vertex form would be done by expanding and completing the square. However, I have come up with an equation that replaces this method, by allowing conversions to be carried out using only substitution and simplification. This method is easier for me, however others may find it more difficult than the traditional method. Therefore, the following can be taken as personal preference, and not an advised replacement, use whichever method works for you.


For the image above, I have created new variables (i and v) for the factored form equation, they represent the x coefficients, and play a key role in the equation below.

equation1.png :If you are having trouble reading the equation, open the image in a new tab.

Once these variables are substituted in, and the equation is simplified, the conversion between factored and vertex form is complete. I have tested this numerous times by subbing in values for i, x, s, v, and t. Each time, both equations return identical values for y. Feel free to test this out, and if you find it more efficient, feel free to use this equation when converting. Below is the standard form conversion equation I have come up with as well. Enjoy!

Last edited:


Forum Staff
Jan 2006
Wellsville, NY
The equation you have given is incorrect. The correct equation would be
\(\displaystyle y = iv \left ( x - \frac{ \left [ \frac{s}{i} + \frac{t}{v} \right ] }{ 2} \right ) ^2 + \left ( - \left [ iv \right ] \left ( \frac{ \left [ \frac{s}{i} + \frac{t}{v} \right ] }{2} \right ) ^2 + \left[ iv \right ]st \right )\)
You didn't distribute the iv properly in the last factor.

Also, as quadratics have roots that can be imaginary I'd skip the use of "i" and use, perhaps, "k."

The same problem occurs in your other form. The last term should be ac.