Solve x^2/x-2+4/2-x=x+2/2x

$$\frac { x ^ { 2 } } { x - 2 } + \frac { 4 } { 2 - x } = \frac { x + 2 } { 2 x }$$

Solve for x

$$x=-2$$

$$x=\frac{1}{2}=0.5$$

Solution Steps

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-2\right), the least common multiple of x-2,2-x,2x.

$$2xx^{2}-2x\times 4=\left(x-2\right)\left(x+2\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Multiply -2 and 4 to get -8.

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

Consider \left(x-2\right)\left(x+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$2x^{3}-8x=x^{2}-4$$

Subtract x^{2} from both sides.

$$2x^{3}-8x-x^{2}=-4$$

Add 4 to both sides.

$$2x^{3}-8x-x^{2}+4=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$2x^{3}-x^{2}-8x+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±2,±4,±1,±\frac{1}{2}$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{3}-x^{2}-8x+4 by x-2 to get 2x^{2}+3x-2. Solve the equation where the result equals to 0.

$$2x^{2}+3x-2=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 2 for a, 3 for b, and -2 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{-3±5}{4}$$

Solve the equation 2x^{2}+3x-2=0 when ± is plus and when ± is minus.

$$x=-2 x=\frac{1}{2}$$

Remove the values that the variable cannot be equal to.

$$x=-2\text{ or }x=\frac{1}{2}$$

List all found solutions.

$$x=2 x=-2 x=\frac{1}{2}$$

Variable x cannot be equal to 2.

$$x=\frac{1}{2} x=-2$$