Solve (x+2)^2+1/4=x/2+(x/2+1/3)^2

$$\frac { ( x + 2 ) ^ { 2 } + 1 } { 4 } = \frac { x } { 2 } + ( \frac { x } { 2 } + \frac { 1 } { 3 } ) ^ { 2 }$$

Solve for x

$$x = -\frac{41}{6} = -6\frac{5}{6} \approx -6.833333333$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4, the least common multiple of 4,2.

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

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

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

Add 4 and 1 to get 5.

$$x^{2}+4x+5=2x+4\left(\frac{x}{2}+\frac{1}{3}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{x}{2} times \frac{3}{3}. Multiply \frac{1}{3} times \frac{2}{2}.

$$x^{2}+4x+5=2x+4\left(\frac{3x}{6}+\frac{2}{6}\right)^{2}$$

Since \frac{3x}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$x^{2}+4x+5=2x+4\times \left(\frac{3x+2}{6}\right)^{2}$$

To raise \frac{3x+2}{6} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+4x+5=2x+4\times \frac{\left(3x+2\right)^{2}}{6^{2}}$$

Express 4\times \frac{\left(3x+2\right)^{2}}{6^{2}} as a single fraction.

$$x^{2}+4x+5=2x+\frac{4\left(3x+2\right)^{2}}{6^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x times \frac{6^{2}}{6^{2}}.

$$x^{2}+4x+5=\frac{2x\times 6^{2}}{6^{2}}+\frac{4\left(3x+2\right)^{2}}{6^{2}}$$

Since \frac{2x\times 6^{2}}{6^{2}} and \frac{4\left(3x+2\right)^{2}}{6^{2}} have the same denominator, add them by adding their numerators.

$$x^{2}+4x+5=\frac{2x\times 6^{2}+4\left(3x+2\right)^{2}}{6^{2}}$$

Do the multiplications in 2x\times 6^{2}+4\left(3x+2\right)^{2}.

$$x^{2}+4x+5=\frac{72x+36x^{2}+48x+16}{6^{2}}$$

Combine like terms in 72x+36x^{2}+48x+16.

$$x^{2}+4x+5=\frac{120x+36x^{2}+16}{6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$x^{2}+4x+5=\frac{120x+36x^{2}+16}{36}$$

Divide each term of 120x+36x^{2}+16 by 36 to get \frac{10}{3}x+x^{2}+\frac{4}{9}.

$$x^{2}+4x+5=\frac{10}{3}x+x^{2}+\frac{4}{9}$$

Subtract \frac{10}{3}x from both sides.

$$x^{2}+4x+5-\frac{10}{3}x=x^{2}+\frac{4}{9}$$

Combine 4x and -\frac{10}{3}x to get \frac{2}{3}x.

$$x^{2}+\frac{2}{3}x+5=x^{2}+\frac{4}{9}$$

Subtract x^{2} from both sides.

$$x^{2}+\frac{2}{3}x+5-x^{2}=\frac{4}{9}$$

Combine x^{2} and -x^{2} to get 0.

$$\frac{2}{3}x+5=\frac{4}{9}$$

Subtract 5 from both sides.

$$\frac{2}{3}x=\frac{4}{9}-5$$

Subtract 5 from \frac{4}{9} to get -\frac{41}{9}.

$$\frac{2}{3}x=-\frac{41}{9}$$

Multiply both sides by \frac{3}{2}, the reciprocal of \frac{2}{3}.

$$x=-\frac{41}{9}\times \frac{3}{2}$$

Multiply -\frac{41}{9} and \frac{3}{2} to get -\frac{41}{6}.

$$x=-\frac{41}{6}$$