Solve 1/3x(x+2)=x+b

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

Solve for b

$$b=\frac{x\left(x-1\right)}{3}$$

Solution Steps

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

$$\frac{1}{3}x^{2}+\frac{2}{3}x=x+b$$

Swap sides so that all variable terms are on the left hand side.

$$x+b=\frac{1}{3}x^{2}+\frac{2}{3}x$$

Subtract x from both sides.

$$b=\frac{1}{3}x^{2}+\frac{2}{3}x-x$$

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

$$b=\frac{1}{3}x^{2}-\frac{1}{3}x$$

Solve for x (complex solution)

$$x=\frac{\sqrt{12b+1}+1}{2}$$

$$x=\frac{-\sqrt{12b+1}+1}{2}$$

Solve for x

$$x=\frac{\sqrt{12b+1}+1}{2}$$

$$x=\frac{-\sqrt{12b+1}+1}{2}\text{, }b\geq -\frac{1}{12}$$