Putting a quadratic into standard form

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Let's start with the function


where a\ne{0} (otherwise we'd have a linear function).

Now we factor out an a:


We want to create a perfect square, by replacing that term


How do we do that?

Well, thinking backwards is often useful. If we consider a perfect square and expand it,


we see that we can rewrite that as


We've solved for the quadratic piece and the linear piece. Now we set this equal to the expression we want to replace.





we can write


from which we arrive at


Finally, following a little simplification, we can write f(x) in standard form as




The former has the advantage of featuring the discriminant, from the quadratic formula. The simplest way of finding the constant is by evaluating


From this form, we can see that the maximum or minimum of the function occurs at


because this is the value at which the squared term is zero. The value of the function at x=-\frac{b}{2a} is


These two special values together are called the vertex: \left(-\frac{b}{2a},\frac{4ac-b^2}{4a}\right).

Example from Stewart


  1. Mathematica File confirming the formula for the y-value of the vertex.
  2. Animation of Standard Form.
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