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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: $$x^2\ +\ 2bx\ =\ c$$ complete the square by adding $b^2$ to both sides: $$(x+b)^2\ =\ b^2\ +\ c$$ and so: $$x+b\ =\pm\sqrt{b^2\ +\ c}$$ or: $$x\ =\ -b\ \pm\sqrt{b^2\ +\ c}$$ FIND THE MINIMUM: $$2(x-a)(x+\sqrt{x^{2}+b^{2}})$$ complete the square once: $$=\ (x-a)^2\ +\ x^2\ -\ a^2\ +\ 2(x-a)\sqrt{x^{2}+b^{2}}$$ then complete the square again: $$=\ (x-a)^2\ +\ 2(x-a)\sqrt{x^2+b^2}\ +\ (x^2+b^2)\ -\ a^2\ -\ b^2$$ $$=\ \left(x\ -\ a\ +\ \sqrt{x^2+b^2}\right)^2\ -\ (a^2\ +\ b^2)$$ so we now only need to find the minimum of $(x\ -\ a\ +\ \sqrt{x^2+b^2})^2$ INTEGRAL: $$\int\frac{dx}{\sqrt{x^2\ +\ 2bx\ +\ c}}$$ $$=\ \int\frac{dx}{\sqrt{(x+b)^2\ -\ b^2\ +\ c}}$$ which is easy to solve by the substitution $y \ =\ x+b$

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 Breakdown Mathematics > Algebra >> Quadratic Forms