Solve (x+1/3)^2+2x-5/2=1/9+x^2-2x-7/6+frac{4+3x}{3}

$$( x + \frac { 1 } { 3 } ) ^ { 2 } + \frac { 2 x - 5 } { 2 } = \frac { 1 } { 9 } + x ^ { 2 } - 2 x - \frac { 7 } { 6 } + \frac { 4 + 3 x } { 3 }$$

Solve for x

$$x=1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 18, the least common multiple of 2,9,6,3.

$$18\left(x+\frac{1}{3}\right)^{2}+9\left(2x-5\right)=2+18x^{2}-36x-21+6\left(4+3x\right)$$

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

$$18\left(x^{2}+\frac{2}{3}x+\frac{1}{9}\right)+9\left(2x-5\right)=2+18x^{2}-36x-21+6\left(4+3x\right)$$

Use the distributive property to multiply 18 by x^{2}+\frac{2}{3}x+\frac{1}{9}.

$$18x^{2}+12x+2+9\left(2x-5\right)=2+18x^{2}-36x-21+6\left(4+3x\right)$$

Use the distributive property to multiply 9 by 2x-5.

$$18x^{2}+12x+2+18x-45=2+18x^{2}-36x-21+6\left(4+3x\right)$$

Combine 12x and 18x to get 30x.

$$18x^{2}+30x+2-45=2+18x^{2}-36x-21+6\left(4+3x\right)$$

Subtract 45 from 2 to get -43.

$$18x^{2}+30x-43=2+18x^{2}-36x-21+6\left(4+3x\right)$$

Subtract 21 from 2 to get -19.

$$18x^{2}+30x-43=-19+18x^{2}-36x+6\left(4+3x\right)$$

Use the distributive property to multiply 6 by 4+3x.

$$18x^{2}+30x-43=-19+18x^{2}-36x+24+18x$$

Add -19 and 24 to get 5.

$$18x^{2}+30x-43=5+18x^{2}-36x+18x$$

Combine -36x and 18x to get -18x.

$$18x^{2}+30x-43=5+18x^{2}-18x$$

Subtract 18x^{2} from both sides.

$$18x^{2}+30x-43-18x^{2}=5-18x$$

Combine 18x^{2} and -18x^{2} to get 0.

$$30x-43=5-18x$$

Add 18x to both sides.

$$30x-43+18x=5$$

Combine 30x and 18x to get 48x.

$$48x-43=5$$

Add 43 to both sides.

$$48x=5+43$$

Add 5 and 43 to get 48.

$$48x=48$$

Divide both sides by 48.

$$x=\frac{48}{48}$$

Divide 48 by 48 to get 1.

$$x=1$$