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Completing the square

A parabola has a minimum or a maximum value. In mathematics, these are both called an extreme value.

 

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Explanation

A parabola is a quadratic function, which you write as

For a > 0, the convex side points downwards and the parabola has the form

If we complete the square, we can easily calculate the coordinates of the extreme value.

 

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Example 1

We want to calculate the lowest value of the function

y = x² + 2x − 15

In the first two terms, we see x² + 2x + ··· which results from (x + 1)² = x² + 2x + 1. If we want to use that, we have to offset the added value 1, so

y = (x² + 2x + 1) − 1 − 15
   = (x + 1)² − 16

A square is always positive, and the lowest value is 0. You get that at x = –1 and then y = −16. You can see that clearly here.

 

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