Solve (x+1)(2/3x-2/3)=(x+4)(8/3-2/3x)

$$( x + 1 ) ( \frac { 2 } { 3 } x - \frac { 2 } { 3 } ) = ( x + 4 ) ( \frac { 8 } { 3 } - \frac { 2 } { 3 } x )$$

Solve for x

$$x = \frac{\sqrt{34}}{2} \approx 2.915475947$$

$$x = -\frac{\sqrt{34}}{2} \approx -2.915475947$$

Steps by Finding Square Root

Use the distributive property to multiply x+1 by \frac{2}{3}x-\frac{2}{3} and combine like terms.

$$\frac{2}{3}x^{2}-\frac{2}{3}=\left(x+4\right)\left(\frac{8}{3}-\frac{2}{3}x\right)$$

Use the distributive property to multiply x+4 by \frac{8}{3}-\frac{2}{3}x and combine like terms.

$$\frac{2}{3}x^{2}-\frac{2}{3}=-\frac{2}{3}x^{2}+\frac{32}{3}$$

Add \frac{2}{3}x^{2} to both sides.

$$\frac{2}{3}x^{2}-\frac{2}{3}+\frac{2}{3}x^{2}=\frac{32}{3}$$

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

$$\frac{4}{3}x^{2}-\frac{2}{3}=\frac{32}{3}$$

Add \frac{2}{3} to both sides.

$$\frac{4}{3}x^{2}=\frac{32}{3}+\frac{2}{3}$$

Add \frac{32}{3} and \frac{2}{3} to get \frac{34}{3}.

$$\frac{4}{3}x^{2}=\frac{34}{3}$$

Multiply both sides by \frac{3}{4}, the reciprocal of \frac{4}{3}.

$$x^{2}=\frac{34}{3}\times \frac{3}{4}$$

Multiply \frac{34}{3} and \frac{3}{4} to get \frac{17}{2}.

$$x^{2}=\frac{17}{2}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{34}}{2} x=-\frac{\sqrt{34}}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+1 by \frac{2}{3}x-\frac{2}{3} and combine like terms.

$$\frac{2}{3}x^{2}-\frac{2}{3}=\left(x+4\right)\left(\frac{8}{3}-\frac{2}{3}x\right)$$

Use the distributive property to multiply x+4 by \frac{8}{3}-\frac{2}{3}x and combine like terms.

$$\frac{2}{3}x^{2}-\frac{2}{3}=-\frac{2}{3}x^{2}+\frac{32}{3}$$

Add \frac{2}{3}x^{2} to both sides.

$$\frac{2}{3}x^{2}-\frac{2}{3}+\frac{2}{3}x^{2}=\frac{32}{3}$$

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

$$\frac{4}{3}x^{2}-\frac{2}{3}=\frac{32}{3}$$

Subtract \frac{32}{3} from both sides.

$$\frac{4}{3}x^{2}-\frac{2}{3}-\frac{32}{3}=0$$

Subtract \frac{32}{3} from -\frac{2}{3} to get -\frac{34}{3}.

$$\frac{4}{3}x^{2}-\frac{34}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{3} for a, 0 for b, and -\frac{34}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{4}{3}\left(-\frac{34}{3}\right)}}{2\times \frac{4}{3}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{4}{3}\left(-\frac{34}{3}\right)}}{2\times \frac{4}{3}}$$

Multiply -4 times \frac{4}{3}.

$$x=\frac{0±\sqrt{-\frac{16}{3}\left(-\frac{34}{3}\right)}}{2\times \frac{4}{3}}$$

Multiply -\frac{16}{3} times -\frac{34}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0±\sqrt{\frac{544}{9}}}{2\times \frac{4}{3}}$$

Take the square root of \frac{544}{9}.

$$x=\frac{0±\frac{4\sqrt{34}}{3}}{2\times \frac{4}{3}}$$

Multiply 2 times \frac{4}{3}.

$$x=\frac{0±\frac{4\sqrt{34}}{3}}{\frac{8}{3}}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{34}}{3}}{\frac{8}{3}} when ± is plus.

$$x=\frac{\sqrt{34}}{2}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{34}}{3}}{\frac{8}{3}} when ± is minus.

$$x=-\frac{\sqrt{34}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{34}}{2} x=-\frac{\sqrt{34}}{2}$$