Solve x^2+2x/x+2-x+3/x=4

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

Solve for x

$$x = \frac{\sqrt{37} + 5}{2} \approx 5.541381265$$

$$x=\frac{5-\sqrt{37}}{2}\approx -0.541381265$$

Solution Steps

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

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

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

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

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$x^{3}+2x^{2}-\left(x^{2}+5x+6\right)=4x\left(x+2\right)$$

To find the opposite of x^{2}+5x+6, find the opposite of each term.

$$x^{3}+2x^{2}-x^{2}-5x-6=4x\left(x+2\right)$$

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

$$x^{3}+x^{2}-5x-6=4x\left(x+2\right)$$

Use the distributive property to multiply 4x by x+2.

$$x^{3}+x^{2}-5x-6=4x^{2}+8x$$

Subtract 4x^{2} from both sides.

$$x^{3}+x^{2}-5x-6-4x^{2}=8x$$

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

$$x^{3}-3x^{2}-5x-6=8x$$

Subtract 8x from both sides.

$$x^{3}-3x^{2}-5x-6-8x=0$$

Combine -5x and -8x to get -13x.

$$x^{3}-3x^{2}-13x-6=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±6,±3,±2,±1$$

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 x^{3}-3x^{2}-13x-6 by x+2 to get x^{2}-5x-3. Solve the equation where the result equals to 0.

$$x^{2}-5x-3=0$$

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 1 for a, -5 for b, and -3 for c in the quadratic formula.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\left(-3\right)}}{2}$$

Do the calculations.

$$x=\frac{5±\sqrt{37}}{2}$$

Solve the equation x^{2}-5x-3=0 when ± is plus and when ± is minus.

$$x=\frac{5-\sqrt{37}}{2} x=\frac{\sqrt{37}+5}{2}$$

Remove the values that the variable cannot be equal to.

$$x\in \emptyset$$

List all found solutions.

$$x=-2 x=\frac{5-\sqrt{37}}{2} x=\frac{\sqrt{37}+5}{2}$$

Variable x cannot be equal to -2.

$$x=\frac{\sqrt{37}+5}{2} x=\frac{5-\sqrt{37}}{2}$$