Exercise 1145Linear algebra › Quadratic formsL1Added 3 Nov 2026

Reducing a quadratic form by completing the square

#reduction#completing square#quadratic form

Statement

Reduce the quadratic form
$Q(x,y) = x^2 + 4xy + 5y^2$
by completing the square.

Hint

Write $Q$ as $(x+2y)^2 + \ldots$ .

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Completing the square.
$Q(x,y) = x^2 + 4xy + 5y^2 = (x+2y)^2 + y^2.$

Verification. $(x+2y)^2+y^2 = x^2+4xy+4y^2+y^2=x^2+4xy+5y^2$ . Correct.

New variables. Let $u=x+2y$ , $v=y$ . Then $Q=u^2+v^2$ .

Conclusion. $Q$ is positive definite (sum of two squares). Its signature is $(2,0,0)$ .