Mathematico | Mathematico

Factorise the expression

x^{2} + y^{2} + z^{2} + 2 x y + 2 y z + 2 z x

Hint

Think of this as a quadratic in terms of $x$ . Then complete the square.

Hint

You can write this in the form

x^{2} + 2 (y + z) x + y^{2} + z^{2} + 2 y z

before completing the square.

Solution

We'll get it in quadratic form in terms of $x$ , then complete the square:

\begin{aligned} x^{2} + y^{2} + z^{2} + 2 x y + 2 y z + 2 z x \\ = & x^{2} + 2 (y + z) x + y^{2} + z^{2} + 2 y z \\ = & (x + y + z)^{2} - (y + z)^{2} + y^{2} + 2 y z + z^{2} \\ = & (x + y + z)^{2} - y^{2} - 2 y z - z^{2} + y^{2} + 2 y z + z^{2} \\ = & (x + y + z)^{2} \end{aligned}