Completing the square

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In algebra, completing the square is a way of rewriting a quadratic polynomial as the sum of a constant and a constant multiple of the square of a first-degree polynomial. Thus one has

and completing the square is the way of filling in the blank between the brackets. Completing the square is used for solving quadratic equations (the proof of the well-known formula for the general solution consists of completing the square). The technique is also used to find the maximum or minimum value of a quadratic function, or in other words, the vertex of a parabola.

The technique relies on the elementary algebraic identity

Concrete examples

We want to fill in this blank:

We write

Now the expression () corresponds to in the elementary identity labeled (*) above. If is and is , then must be 7. Therefore must be . So we continue:

Now we have added inside the parentheses, and compensated (thus justifying the "=") by subtracting outside the parentheses. The expression inside the parentheses is now , and by the elementary identity labeled (*) above, it is therefore equal to , i.e. to . So now we have

Thus we have the equality

More abstractly

It is possible to give a closed formula for the completion in terms of the coefficients a, b and c. Namely,

where stands for the well-known discriminant of the polynomial, that is .

Indeed, we have

The last expression inside parentheses above corresponds to in the identity labelled (*) above. We need to add the third term, :