What is expand (x^2+y^2+z^2)/2-((x+y+z)^2)/6 ?

The solution to expand (x^2+y^2+z^2)/2-((x+y+z)^2)/6 is (x^2)/3-(xy)/3-(xz)/3+(y^2)/3+(z^2)/3-(yz)/3

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expand (x^2+y^2+z^2)/2-((x+y+z)^2)/6

expand

\frac{x ^{2} + y ^{2} + z ^{2}}{2} - \frac{( x + y + z ) ^{2}}{6}

\frac{x ^{2}}{3} - \frac{x y}{3} - \frac{x z}{3} + \frac{y ^{2}}{3} + \frac{z ^{2}}{3} - \frac{yz}{3}

Solution steps

\frac{x ^{2} + y ^{2} + z ^{2}}{2} - \frac{( x + y + z ) ^{2}}{6}

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

= \frac{x ^{2} + y ^{2} + z ^{2}}{2} - \frac{x ^{2} + 2 x y + 2 x z + y ^{2} + 2 yz + z ^{2}}{6}

Least Common Multiplier of

2, 6 : 6

Adjust Fractions based on the LCM

= \frac{( x ^{2} + y ^{2} + z ^{2} ) \cdot 3}{6} - \frac{2 x y + 2 x z + x ^{2} + y ^{2} + 2 yz + z ^{2}}{6}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{( x ^{2} + y ^{2} + z ^{2} ) \cdot 3 - ( 2 x y + 2 x z + x ^{2} + y ^{2} + 2 yz + z ^{2} )}{6}

Expand

(x^{2} + y^{2} + z^{2}) \cdot 3 - (2 x y + 2 x z + x^{2} + y^{2} + 2 yz + z^{2}) : 2 x^{2} - 2 x y - 2 x z + 2 y^{2} + 2 z^{2} - 2 yz

= \frac{2 x ^{2} - 2 x y - 2 x z + 2 y ^{2} + 2 z ^{2} - 2 yz}{6}

Factor out common term

2

= \frac{2 ( x ^{2} - x y - x z + y ^{2} + z ^{2} - yz )}{6}

Cancel the common factor:

2

= \frac{x ^{2} - x y - x z + y ^{2} + z ^{2} - yz}{3}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{x ^{2} - x y - x z + y ^{2} + z ^{2} - yz}{3} = \frac{x ^{2}}{3} - \frac{x y}{3} - \frac{x z}{3} + \frac{y ^{2}}{3} + \frac{z ^{2}}{3} - \frac{yz}{3}

= \frac{x ^{2}}{3} - \frac{x y}{3} - \frac{x z}{3} + \frac{y ^{2}}{3} + \frac{z ^{2}}{3} - \frac{yz}{3}

\frac{sec ^{2} ( x )}{( 1 + tan ( x ) ) ^{3}}

\frac{4}{s ^{3} ( s ^{2} - s )}

\frac{8 ( 154 - 2 )}{5}

\frac{x ^{2} + 4 x + 7}{( x + 2 ) ^{2}}

\frac{( x + 3 ) ( x + 2 )}{x - 3}

What is expand (x^2+y^2+z^2)/2-((x+y+z)^2)/6 ?
The solution to expand (x^2+y^2+z^2)/2-((x+y+z)^2)/6 is (x^2)/3-(xy)/3-(xz)/3+(y^2)/3+(z^2)/3-(yz)/3