Solve (x-2)^2/3-5x+1/2=(x-2)(x+2)/3

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

Solve for x

$$x=\frac{13}{23}\approx 0.565217391$$

Steps for Solving Linear Equation

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

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

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

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

Use the distributive property to multiply 2 by x^{2}-4x+4.

$$2x^{2}-8x+8-3\left(5x+1\right)=2\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply -3 by 5x+1.

$$2x^{2}-8x+8-15x-3=2\left(x-2\right)\left(x+2\right)$$

Combine -8x and -15x to get -23x.

$$2x^{2}-23x+8-3=2\left(x-2\right)\left(x+2\right)$$

Subtract 3 from 8 to get 5.

$$2x^{2}-23x+5=2\left(x-2\right)\left(x+2\right)$$

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

$$2x^{2}-23x+5=\left(2x-4\right)\left(x+2\right)$$

Use the distributive property to multiply 2x-4 by x+2 and combine like terms.

$$2x^{2}-23x+5=2x^{2}-8$$

Subtract 2x^{2} from both sides.

$$2x^{2}-23x+5-2x^{2}=-8$$

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

$$-23x+5=-8$$

Subtract 5 from both sides.

$$-23x=-8-5$$

Subtract 5 from -8 to get -13.

$$-23x=-13$$

Divide both sides by -23.

$$x=\frac{-13}{-23}$$

Fraction \frac{-13}{-23} can be simplified to \frac{13}{23} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{13}{23}$$