Verify that $(x + y + z)^{2}$ = $x^{2} + y^{2} + z^{2} + 2 x y + 2 y z + 2 z x$ .

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Solution

Let consider k = (y + z)

$(x + y + z)^{2}$ = $(x + k)^{2}$
= $x^{2} + k^{2} + 2 x k$ ….(eq1)

Now, put the value of k in above equation

$(x + y + z)^{2}$ = $x^{2} + k^{2} + 2 x k$
= $x^{2} + (y + z)^{2} + 2 (y + z) x$
= $x^{2} + y^{2} + 2 y z + z^{2} + 2 y x + 2 z x$
= $x^{2} + y^{2} + z^{2} + 2 y z + 2 y x + 2 z x$
or $x^{2} + y^{2} + z^{2} + 2 x y + 2 y z + 2 z x$

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