Solve x^2+{left(1/3x^2-4/3xright)}^2

$${ x }^{ 2 } + { \left( \frac{ 1 }{ 3 } { x }^{ 2 } - \frac{ 4 }{ 3 } x \right) }^{ 2 }$$

Evaluate

$$\frac{x^{2}\left(x^{2}-8x+25\right)}{9}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{1}{3}x^{2}-\frac{4}{3}x\right)^{2}.

$$x^{2}+\frac{1}{9}\left(x^{2}\right)^{2}-\frac{8}{9}x^{2}x+\frac{16}{9}x^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{2}+\frac{1}{9}x^{4}-\frac{8}{9}x^{2}x+\frac{16}{9}x^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{2}+\frac{1}{9}x^{4}-\frac{8}{9}x^{3}+\frac{16}{9}x^{2}$$

Combine x^{2} and \frac{16}{9}x^{2} to get \frac{25}{9}x^{2}.

$$\frac{25}{9}x^{2}+\frac{1}{9}x^{4}-\frac{8}{9}x^{3}$$

Expand

$$\frac{x^{4}-8x^{3}+25x^{2}}{9}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{1}{3}x^{2}-\frac{4}{3}x\right)^{2}.

$$x^{2}+\frac{1}{9}\left(x^{2}\right)^{2}-\frac{8}{9}x^{2}x+\frac{16}{9}x^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{2}+\frac{1}{9}x^{4}-\frac{8}{9}x^{2}x+\frac{16}{9}x^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{2}+\frac{1}{9}x^{4}-\frac{8}{9}x^{3}+\frac{16}{9}x^{2}$$

Combine x^{2} and \frac{16}{9}x^{2} to get \frac{25}{9}x^{2}.

$$\frac{25}{9}x^{2}+\frac{1}{9}x^{4}-\frac{8}{9}x^{3}$$