Solve x^2+{left(-1/8x^2+8-6right)}^2

$${ x }^{ 2 } + { \left(- \frac{ 1 }{ 8 } { x }^{ 2 } +8-6 \right) }^{ 2 }$$

Evaluate

$$\frac{\left(x^{2}+16\right)^{2}}{64}$$

Solution Steps

Subtract 6 from 8 to get 2.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{1}{8}x^{2}+2\right)^{2}.

$$x^{2}+\frac{1}{64}\left(x^{2}\right)^{2}-\frac{1}{2}x^{2}+4$$

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

$$x^{2}+\frac{1}{64}x^{4}-\frac{1}{2}x^{2}+4$$

Combine x^{2} and -\frac{1}{2}x^{2} to get \frac{1}{2}x^{2}.

$$\frac{1}{2}x^{2}+\frac{1}{64}x^{4}+4$$

Expand

$$\frac{x^{4}}{64}+\frac{x^{2}}{2}+4$$

Solution Steps

Subtract 6 from 8 to get 2.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{1}{8}x^{2}+2\right)^{2}.

$$x^{2}+\frac{1}{64}\left(x^{2}\right)^{2}-\frac{1}{2}x^{2}+4$$

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

$$x^{2}+\frac{1}{64}x^{4}-\frac{1}{2}x^{2}+4$$

Combine x^{2} and -\frac{1}{2}x^{2} to get \frac{1}{2}x^{2}.

$$\frac{1}{2}x^{2}+\frac{1}{64}x^{4}+4$$