Solve sum(x-1/2)^2-(x+1/2)^2=-1/3(1-x)+1/2(x+2)

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

Solve for Σ

$$Σ=\frac{3x^{2}+16x+11}{3\left(x-1\right)^{2}}$$

$$x\neq 1$$

Steps for Solving Linear Equation

To raise \frac{x-1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Express Σ\times \frac{\left(x-1\right)^{2}}{2^{2}} as a single fraction.

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

To raise \frac{x+1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

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

Since \frac{Σ\left(x-1\right)^{2}}{4} and \frac{x^{2}+2x+1}{4} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in Σ\left(x-1\right)^{2}-\left(x^{2}+2x+1\right).

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

Use the distributive property to multiply -\frac{1}{3} by 1-x.

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

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

$$\frac{Σx^{2}-2Σx+Σ-x^{2}-2x-1}{4}=-\frac{1}{3}+\frac{1}{3}x+\frac{1}{2}x+1$$

Combine \frac{1}{3}x and \frac{1}{2}x to get \frac{5}{6}x.

$$\frac{Σx^{2}-2Σx+Σ-x^{2}-2x-1}{4}=-\frac{1}{3}+\frac{5}{6}x+1$$

Add -\frac{1}{3} and 1 to get \frac{2}{3}.

$$\frac{Σx^{2}-2Σx+Σ-x^{2}-2x-1}{4}=\frac{2}{3}+\frac{5}{6}x$$

Multiply both sides of the equation by 12, the least common multiple of 4,3,6.

$$3\left(Σx^{2}-2Σx+Σ-x^{2}-2x-1\right)=8+10x$$

Use the distributive property to multiply 3 by Σx^{2}-2Σx+Σ-x^{2}-2x-1.

$$3Σx^{2}-6Σx+3Σ-3x^{2}-6x-3=8+10x$$

Add 3x^{2} to both sides.

$$3Σx^{2}-6Σx+3Σ-6x-3=8+10x+3x^{2}$$

Add 6x to both sides.

$$3Σx^{2}-6Σx+3Σ-3=8+10x+3x^{2}+6x$$

Combine 10x and 6x to get 16x.

$$3Σx^{2}-6Σx+3Σ-3=8+16x+3x^{2}$$

Add 3 to both sides.

$$3Σx^{2}-6Σx+3Σ=8+16x+3x^{2}+3$$

Add 8 and 3 to get 11.

$$3Σx^{2}-6Σx+3Σ=11+16x+3x^{2}$$

Combine all terms containing Σ.

$$\left(3x^{2}-6x+3\right)Σ=11+16x+3x^{2}$$

The equation is in standard form.

$$\left(3x^{2}-6x+3\right)Σ=3x^{2}+16x+11$$

Divide both sides by 3x^{2}-6x+3.

$$\frac{\left(3x^{2}-6x+3\right)Σ}{3x^{2}-6x+3}=\frac{3x^{2}+16x+11}{3x^{2}-6x+3}$$

Dividing by 3x^{2}-6x+3 undoes the multiplication by 3x^{2}-6x+3.

$$Σ=\frac{3x^{2}+16x+11}{3x^{2}-6x+3}$$

Divide 11+16x+3x^{2} by 3x^{2}-6x+3.

$$Σ=\frac{3x^{2}+16x+11}{3\left(x-1\right)^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{3Σ+\sqrt{90Σ+31}+8}{3\left(Σ-1\right)}\text{; }x=\frac{3Σ-\sqrt{90Σ+31}+8}{3\left(Σ-1\right)}\text{, }&Σ\neq 1\\x=-\frac{4}{11}\approx -0.363636364\text{, }&Σ=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{3Σ+\sqrt{90Σ+31}+8}{3\left(Σ-1\right)}\text{; }x=\frac{3Σ-\sqrt{90Σ+31}+8}{3\left(Σ-1\right)}\text{, }&Σ\neq 1\text{ and }Σ\geq -\frac{31}{90}\\x=-\frac{4}{11}\approx -0.363636364\text{, }&Σ=1\end{matrix}\right.$$