Solve (x+1/6)^2-2/3x(x+1/6)^2

$$( x + \frac { 1 } { 6 } ) ^ { 2 } - \frac { 2 } { 3 } x ( x + \frac { 1 } { 6 } ) ^ { 2 }$$

Evaluate

$$\frac{\left(3-2x\right)\left(6x+1\right)^{2}}{108}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\frac{1}{6}\right)^{2}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x\left(x+\frac{1}{6}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\frac{1}{6}\right)^{2}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x\left(x^{2}+\frac{1}{3}x+\frac{1}{36}\right)$$

Use the distributive property to multiply -\frac{2}{3}x by x^{2}+\frac{1}{3}x+\frac{1}{36}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x^{3}-\frac{2}{9}x^{2}-\frac{1}{54}x$$

Combine x^{2} and -\frac{2}{9}x^{2} to get \frac{7}{9}x^{2}.

$$\frac{7}{9}x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x^{3}-\frac{1}{54}x$$

Combine \frac{1}{3}x and -\frac{1}{54}x to get \frac{17}{54}x.

$$\frac{7}{9}x^{2}+\frac{17}{54}x+\frac{1}{36}-\frac{2}{3}x^{3}$$

Expand

$$-\frac{2x^{3}}{3}+\frac{7x^{2}}{9}+\frac{17x}{54}+\frac{1}{36}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\frac{1}{6}\right)^{2}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x\left(x+\frac{1}{6}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\frac{1}{6}\right)^{2}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x\left(x^{2}+\frac{1}{3}x+\frac{1}{36}\right)$$

Use the distributive property to multiply -\frac{2}{3}x by x^{2}+\frac{1}{3}x+\frac{1}{36}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x^{3}-\frac{2}{9}x^{2}-\frac{1}{54}x$$

Combine x^{2} and -\frac{2}{9}x^{2} to get \frac{7}{9}x^{2}.

$$\frac{7}{9}x^{2}+\frac{1}{3}x+\frac{1}{36}-\frac{2}{3}x^{3}-\frac{1}{54}x$$

Combine \frac{1}{3}x and -\frac{1}{54}x to get \frac{17}{54}x.

$$\frac{7}{9}x^{2}+\frac{17}{54}x+\frac{1}{36}-\frac{2}{3}x^{3}$$