Solve (x+6)^{2}+(x^2/4+9)^2=(x^2/4-9)^2

$$( x + 6 ) ^ { 2 } + ( \frac { x ^ { 2 } } { 4 } + 9 ) ^ { 2 } = ( \frac { x ^ { 2 } } { 4 } - 9 ) ^ { 2 }$$

Solve for x (complex solution)

$$x=-\frac{3}{5}+\frac{9}{5}i=-0.6+1.8i$$

$$x=-\frac{3}{5}-\frac{9}{5}i=-0.6-1.8i$$

Steps Using the Quadratic Formula

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

$$x^{2}+12x+36+\left(\frac{x^{2}}{4}+9\right)^{2}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{x^{2}}{4}+9\right)^{2}.

$$x^{2}+12x+36+\left(\frac{x^{2}}{4}\right)^{2}+18\times \frac{x^{2}}{4}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To raise \frac{x^{2}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{4^{2}}+18\times \frac{x^{2}}{4}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Express 18\times \frac{x^{2}}{4} as a single fraction.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{18x^{2}}{4}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 4 is 16. Multiply \frac{18x^{2}}{4} times \frac{4}{4}.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{16}+\frac{4\times 18x^{2}}{16}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{\left(x^{2}\right)^{2}}{16} and \frac{4\times 18x^{2}}{16} have the same denominator, add them by adding their numerators.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+4\times 18x^{2}}{16}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 81 times \frac{4^{2}}{4^{2}}.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{18x^{2}}{4}+\frac{81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{\left(x^{2}\right)^{2}}{4^{2}} and \frac{81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{18x^{2}}{4}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Divide 18x^{2} by 4 to get \frac{9}{2}x^{2}.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{9}{2}x^{2}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Combine x^{2} and \frac{9}{2}x^{2} to get \frac{11}{2}x^{2}.

$$\frac{11}{2}x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12x+36 times \frac{4^{2}}{4^{2}}.

$$\frac{11}{2}x^{2}+\frac{\left(12x+36\right)\times 4^{2}}{4^{2}}+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{\left(12x+36\right)\times 4^{2}}{4^{2}} and \frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+\frac{\left(12x+36\right)\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36 times \frac{4^{2}}{4^{2}}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}}{4^{2}}+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{36\times 4^{2}}{4^{2}} and \frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{x^{2}}{4}-9\right)^{2}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}\right)^{2}-18\times \frac{x^{2}}{4}+81$$

To raise \frac{x^{2}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{4^{2}}-18\times \frac{x^{2}}{4}+81$$

Express -18\times \frac{x^{2}}{4} as a single fraction.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{-18x^{2}}{4}+81$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 4 is 16. Multiply \frac{-18x^{2}}{4} times \frac{4}{4}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{16}+\frac{4\left(-1\right)\times 18x^{2}}{16}+81$$

Since \frac{\left(x^{2}\right)^{2}}{16} and \frac{4\left(-1\right)\times 18x^{2}}{16} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}+4\left(-1\right)\times 18x^{2}}{16}+81$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 81 times \frac{4^{2}}{4^{2}}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{-18x^{2}}{4}+\frac{81\times 4^{2}}{4^{2}}$$

Since \frac{\left(x^{2}\right)^{2}}{4^{2}} and \frac{81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

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

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+x^{4}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

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

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+x^{4}+81\times 4^{2}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 16+x^{4}+81\times 4^{2}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Multiply 36 and 16 to get 576.

$$\frac{11}{2}x^{2}+12x+\frac{576+x^{4}+81\times 4^{2}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+\frac{576+x^{4}+81\times 16}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Multiply 81 and 16 to get 1296.

$$\frac{11}{2}x^{2}+12x+\frac{576+x^{4}+1296}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Add 576 and 1296 to get 1872.

$$\frac{11}{2}x^{2}+12x+\frac{1872+x^{4}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+\frac{1872+x^{4}}{16}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Divide each term of 1872+x^{4} by 16 to get 117+\frac{1}{16}x^{4}.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+81\times 16}{4^{2}}+\frac{-18x^{2}}{4}$$

Multiply 81 and 16 to get 1296.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+1296}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+1296}{16}+\frac{-18x^{2}}{4}$$

Divide each term of x^{4}+1296 by 16 to get \frac{1}{16}x^{4}+81.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{1}{16}x^{4}+81+\frac{-18x^{2}}{4}$$

Divide -18x^{2} by 4 to get -\frac{9}{2}x^{2}.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{1}{16}x^{4}+81-\frac{9}{2}x^{2}$$

Subtract \frac{1}{16}x^{4} from both sides.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}-\frac{1}{16}x^{4}=81-\frac{9}{2}x^{2}$$

Combine \frac{1}{16}x^{4} and -\frac{1}{16}x^{4} to get 0.

$$\frac{11}{2}x^{2}+12x+117=81-\frac{9}{2}x^{2}$$

Subtract 81 from both sides.

$$\frac{11}{2}x^{2}+12x+117-81=-\frac{9}{2}x^{2}$$

Subtract 81 from 117 to get 36.

$$\frac{11}{2}x^{2}+12x+36=-\frac{9}{2}x^{2}$$

Add \frac{9}{2}x^{2} to both sides.

$$\frac{11}{2}x^{2}+12x+36+\frac{9}{2}x^{2}=0$$

Combine \frac{11}{2}x^{2} and \frac{9}{2}x^{2} to get 10x^{2}.

$$10x^{2}+12x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 12 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 10\times 36}}{2\times 10}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 10\times 36}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-12±\sqrt{144-40\times 36}}{2\times 10}$$

Multiply -40 times 36.

$$x=\frac{-12±\sqrt{144-1440}}{2\times 10}$$

Add 144 to -1440.

$$x=\frac{-12±\sqrt{-1296}}{2\times 10}$$

Take the square root of -1296.

$$x=\frac{-12±36i}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-12±36i}{20}$$

Now solve the equation x=\frac{-12±36i}{20} when ± is plus. Add -12 to 36i.

$$x=\frac{-12+36i}{20}$$

Divide -12+36i by 20.

$$x=-\frac{3}{5}+\frac{9}{5}i$$

Now solve the equation x=\frac{-12±36i}{20} when ± is minus. Subtract 36i from -12.

$$x=\frac{-12-36i}{20}$$

Divide -12-36i by 20.

$$x=-\frac{3}{5}-\frac{9}{5}i$$

The equation is now solved.

$$x=-\frac{3}{5}+\frac{9}{5}i x=-\frac{3}{5}-\frac{9}{5}i$$

Steps for Completing the Square

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

$$x^{2}+12x+36+\left(\frac{x^{2}}{4}+9\right)^{2}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{x^{2}}{4}+9\right)^{2}.

$$x^{2}+12x+36+\left(\frac{x^{2}}{4}\right)^{2}+18\times \frac{x^{2}}{4}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To raise \frac{x^{2}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{4^{2}}+18\times \frac{x^{2}}{4}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Express 18\times \frac{x^{2}}{4} as a single fraction.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{18x^{2}}{4}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 4 is 16. Multiply \frac{18x^{2}}{4} times \frac{4}{4}.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{16}+\frac{4\times 18x^{2}}{16}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{\left(x^{2}\right)^{2}}{16} and \frac{4\times 18x^{2}}{16} have the same denominator, add them by adding their numerators.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+4\times 18x^{2}}{16}+81=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 81 times \frac{4^{2}}{4^{2}}.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{18x^{2}}{4}+\frac{81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{\left(x^{2}\right)^{2}}{4^{2}} and \frac{81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{18x^{2}}{4}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Divide 18x^{2} by 4 to get \frac{9}{2}x^{2}.

$$x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{9}{2}x^{2}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Combine x^{2} and \frac{9}{2}x^{2} to get \frac{11}{2}x^{2}.

$$\frac{11}{2}x^{2}+12x+36+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12x+36 times \frac{4^{2}}{4^{2}}.

$$\frac{11}{2}x^{2}+\frac{\left(12x+36\right)\times 4^{2}}{4^{2}}+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{\left(12x+36\right)\times 4^{2}}{4^{2}} and \frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+\frac{\left(12x+36\right)\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36 times \frac{4^{2}}{4^{2}}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}}{4^{2}}+\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Since \frac{36\times 4^{2}}{4^{2}} and \frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}-9\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{x^{2}}{4}-9\right)^{2}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\left(\frac{x^{2}}{4}\right)^{2}-18\times \frac{x^{2}}{4}+81$$

To raise \frac{x^{2}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{4^{2}}-18\times \frac{x^{2}}{4}+81$$

Express -18\times \frac{x^{2}}{4} as a single fraction.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{-18x^{2}}{4}+81$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 4 is 16. Multiply \frac{-18x^{2}}{4} times \frac{4}{4}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{16}+\frac{4\left(-1\right)\times 18x^{2}}{16}+81$$

Since \frac{\left(x^{2}\right)^{2}}{16} and \frac{4\left(-1\right)\times 18x^{2}}{16} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}+4\left(-1\right)\times 18x^{2}}{16}+81$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 81 times \frac{4^{2}}{4^{2}}.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}}{4^{2}}+\frac{-18x^{2}}{4}+\frac{81\times 4^{2}}{4^{2}}$$

Since \frac{\left(x^{2}\right)^{2}}{4^{2}} and \frac{81\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

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

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+x^{4}+81\times 4^{2}}{4^{2}}=\frac{\left(x^{2}\right)^{2}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

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

$$\frac{11}{2}x^{2}+12x+\frac{36\times 4^{2}+x^{4}+81\times 4^{2}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+\frac{36\times 16+x^{4}+81\times 4^{2}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Multiply 36 and 16 to get 576.

$$\frac{11}{2}x^{2}+12x+\frac{576+x^{4}+81\times 4^{2}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+\frac{576+x^{4}+81\times 16}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Multiply 81 and 16 to get 1296.

$$\frac{11}{2}x^{2}+12x+\frac{576+x^{4}+1296}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Add 576 and 1296 to get 1872.

$$\frac{11}{2}x^{2}+12x+\frac{1872+x^{4}}{4^{2}}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+\frac{1872+x^{4}}{16}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Divide each term of 1872+x^{4} by 16 to get 117+\frac{1}{16}x^{4}.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+81\times 4^{2}}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+81\times 16}{4^{2}}+\frac{-18x^{2}}{4}$$

Multiply 81 and 16 to get 1296.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+1296}{4^{2}}+\frac{-18x^{2}}{4}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{x^{4}+1296}{16}+\frac{-18x^{2}}{4}$$

Divide each term of x^{4}+1296 by 16 to get \frac{1}{16}x^{4}+81.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{1}{16}x^{4}+81+\frac{-18x^{2}}{4}$$

Divide -18x^{2} by 4 to get -\frac{9}{2}x^{2}.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}=\frac{1}{16}x^{4}+81-\frac{9}{2}x^{2}$$

Subtract \frac{1}{16}x^{4} from both sides.

$$\frac{11}{2}x^{2}+12x+117+\frac{1}{16}x^{4}-\frac{1}{16}x^{4}=81-\frac{9}{2}x^{2}$$

Combine \frac{1}{16}x^{4} and -\frac{1}{16}x^{4} to get 0.

$$\frac{11}{2}x^{2}+12x+117=81-\frac{9}{2}x^{2}$$

Add \frac{9}{2}x^{2} to both sides.

$$\frac{11}{2}x^{2}+12x+117+\frac{9}{2}x^{2}=81$$

Combine \frac{11}{2}x^{2} and \frac{9}{2}x^{2} to get 10x^{2}.

$$10x^{2}+12x+117=81$$

Subtract 117 from both sides.

$$10x^{2}+12x=81-117$$

Subtract 117 from 81 to get -36.

$$10x^{2}+12x=-36$$

Divide both sides by 10.

$$\frac{10x^{2}+12x}{10}=-\frac{36}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{12}{10}x=-\frac{36}{10}$$

Reduce the fraction \frac{12}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{6}{5}x=-\frac{36}{10}$$

Reduce the fraction \frac{-36}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{6}{5}x=-\frac{18}{5}$$

Divide \frac{6}{5}, the coefficient of the x term, by 2 to get \frac{3}{5}. Then add the square of \frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=-\frac{18}{5}+\left(\frac{3}{5}\right)^{2}$$

Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=-\frac{18}{5}+\frac{9}{25}$$

Add -\frac{18}{5} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=-\frac{81}{25}$$

Factor x^{2}+\frac{6}{5}x+\frac{9}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{5}\right)^{2}=-\frac{81}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{5}\right)^{2}}=\sqrt{-\frac{81}{25}}$$

Simplify.

$$x+\frac{3}{5}=\frac{9}{5}i x+\frac{3}{5}=-\frac{9}{5}i$$

Subtract \frac{3}{5} from both sides of the equation.

$$x=-\frac{3}{5}+\frac{9}{5}i x=-\frac{3}{5}-\frac{9}{5}i$$