What is expand ((x(x+1))/2)^2 ?

The solution to expand ((x(x+1))/2)^2 is (x^4)/4+(x^3)/2+(x^2)/4

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expand ((x(x+1))/2)^2

expand

(\frac{x ( x + 1 )}{2})^{2}

\frac{x ^{4}}{4} + \frac{x ^{3}}{2} + \frac{x ^{2}}{4}

Solution steps

(\frac{x ( x + 1 )}{2})^{2}

\frac{x ( x + 1 )}{2} = \frac{x ^{2} + x}{2}

= (\frac{x ^{2} + x}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( x ^{2} + x ) ^{2}}{2 ^{2}}

(x^{2} + x)^{2} = x^{4} + 2 x^{3} + x^{2}

= \frac{x ^{4} + 2 x ^{3} + x ^{2}}{2 ^{2}}

2^{2} = 4

= \frac{x ^{4} + 2 x ^{3} + x ^{2}}{4}

Apply the fraction rule:

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

\frac{x ^{4} + 2 x ^{3} + x ^{2}}{4} = \frac{x ^{4}}{4} + \frac{2 x ^{3}}{4} + \frac{x ^{2}}{4}

= \frac{x ^{4}}{4} + \frac{2 x ^{3}}{4} + \frac{x ^{2}}{4}

Cancel

\frac{2 x ^{3}}{4} : \frac{x ^{3}}{2}

= \frac{x ^{4}}{4} + \frac{x ^{3}}{2} + \frac{x ^{2}}{4}

\frac{\partial}{\partial x} ((z) (5 x y - 4 x + 5 y^{3}))

\frac{2 x - 1}{( x - 1 ) ^{2}}

(\frac{3}{4} x + 2)^{2}

e^{(x + i y)^{2}}

(- e^{x} sin (x) + e^{x} cos (x))^{2}

What is expand ((x(x+1))/2)^2 ?
The solution to expand ((x(x+1))/2)^2 is (x^4)/4+(x^3)/2+(x^2)/4