Science Term: complete the square

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complete the square

Definition/Summary

An expression (usually a quadratic) can often be simplified by adding a constant to it to make a perfect square.

This may make the solution of equations or of integrals much easier.

Equations

QUADRATIC EQUATION:

[tex]x^2\ +\ 2bx\ =\ c[/tex]

complete the square by adding [itex]b^2[/itex] to both sides:

[tex](x+b)^2\ =\ b^2\ +\ c[/tex]

and so:

[tex]x+b\ =\pm\sqrt{b^2\ +\ c}[/tex]

or:

[tex]x\ =\ -b\ \pm\sqrt{b^2\ +\ c}[/tex]

FIND THE MINIMUM:

[tex]2(x-a)(x+\sqrt{x^{2}+b^{2}})[/tex]

complete the square once:

[tex]=\ (x-a)^2\ +\ x^2\ -\ a^2\ +\ 2(x-a)\sqrt{x^{2}+b^{2}}[/tex]

then complete the square again:

[tex]=\ (x-a)^2\ +\ 2(x-a)\sqrt{x^2+b^2}\ +\ (x^2+b^2)\ -\ a^2\ -\ b^2[/tex]

[tex]=\ \left(x\ -\ a\ +\ \sqrt{x^2+b^2}\right)^2\ -\ (a^2\ +\ b^2)[/tex]

so we now only need to find the minimum of [itex](x\ -\ a\ +\ \sqrt{x^2+b^2})^2[/itex]

INTEGRAL:

[tex]\int\frac{dx}{\sqrt{x^2\ +\ 2bx\ +\ c}}[/tex]

[tex]=\ \int\frac{dx}{\sqrt{(x+b)^2\ -\ b^2\ +\ c}}[/tex]

which is easy to solve by the substitution [itex]y \ =\ x+b[/itex]

Scientists

Recent forum threads on complete the square

Breakdown

Mathematics
> Algebra
>> Quadratic Forms