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

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

Solve for x

$$x = \frac{\sqrt{57}}{3} \approx 2.516611478$$

$$x = -\frac{\sqrt{57}}{3} \approx -2.516611478$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x+3,x-1,\left(x+3\right)\left(x-3\right).

$$\left(x-3\right)\left(x-1\right)\left(x+2\right)-\left(x^{2}-9\right)\left(x+1\right)=\left(x-1\right)\times 4$$

Use the distributive property to multiply x-3 by x-1 and combine like terms.

$$\left(x^{2}-4x+3\right)\left(x+2\right)-\left(x^{2}-9\right)\left(x+1\right)=\left(x-1\right)\times 4$$

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

$$x^{3}-2x^{2}-5x+6-\left(x^{2}-9\right)\left(x+1\right)=\left(x-1\right)\times 4$$

Use the distributive property to multiply x^{2}-9 by x+1.

$$x^{3}-2x^{2}-5x+6-\left(x^{3}+x^{2}-9x-9\right)=\left(x-1\right)\times 4$$

To find the opposite of x^{3}+x^{2}-9x-9, find the opposite of each term.

$$x^{3}-2x^{2}-5x+6-x^{3}-x^{2}+9x+9=\left(x-1\right)\times 4$$

Combine x^{3} and -x^{3} to get 0.

$$-2x^{2}-5x+6-x^{2}+9x+9=\left(x-1\right)\times 4$$

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

$$-3x^{2}-5x+6+9x+9=\left(x-1\right)\times 4$$

Combine -5x and 9x to get 4x.

$$-3x^{2}+4x+6+9=\left(x-1\right)\times 4$$

Add 6 and 9 to get 15.

$$-3x^{2}+4x+15=\left(x-1\right)\times 4$$

Use the distributive property to multiply x-1 by 4.

$$-3x^{2}+4x+15=4x-4$$

Subtract 4x from both sides.

$$-3x^{2}+4x+15-4x=-4$$

Combine 4x and -4x to get 0.

$$-3x^{2}+15=-4$$

Subtract 15 from both sides.

$$-3x^{2}=-4-15$$

Subtract 15 from -4 to get -19.

$$-3x^{2}=-19$$

Divide both sides by -3.

$$x^{2}=\frac{-19}{-3}$$

Fraction \frac{-19}{-3} can be simplified to \frac{19}{3} by removing the negative sign from both the numerator and the denominator.

$$x^{2}=\frac{19}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{57}}{3} x=-\frac{\sqrt{57}}{3}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x+3,x-1,\left(x+3\right)\left(x-3\right).

$$\left(x-3\right)\left(x-1\right)\left(x+2\right)-\left(x^{2}-9\right)\left(x+1\right)=\left(x-1\right)\times 4$$

Use the distributive property to multiply x-3 by x-1 and combine like terms.

$$\left(x^{2}-4x+3\right)\left(x+2\right)-\left(x^{2}-9\right)\left(x+1\right)=\left(x-1\right)\times 4$$

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

$$x^{3}-2x^{2}-5x+6-\left(x^{2}-9\right)\left(x+1\right)=\left(x-1\right)\times 4$$

Use the distributive property to multiply x^{2}-9 by x+1.

$$x^{3}-2x^{2}-5x+6-\left(x^{3}+x^{2}-9x-9\right)=\left(x-1\right)\times 4$$

To find the opposite of x^{3}+x^{2}-9x-9, find the opposite of each term.

$$x^{3}-2x^{2}-5x+6-x^{3}-x^{2}+9x+9=\left(x-1\right)\times 4$$

Combine x^{3} and -x^{3} to get 0.

$$-2x^{2}-5x+6-x^{2}+9x+9=\left(x-1\right)\times 4$$

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

$$-3x^{2}-5x+6+9x+9=\left(x-1\right)\times 4$$

Combine -5x and 9x to get 4x.

$$-3x^{2}+4x+6+9=\left(x-1\right)\times 4$$

Add 6 and 9 to get 15.

$$-3x^{2}+4x+15=\left(x-1\right)\times 4$$

Use the distributive property to multiply x-1 by 4.

$$-3x^{2}+4x+15=4x-4$$

Subtract 4x from both sides.

$$-3x^{2}+4x+15-4x=-4$$

Combine 4x and -4x to get 0.

$$-3x^{2}+15=-4$$

Add 4 to both sides.

$$-3x^{2}+15+4=0$$

Add 15 and 4 to get 19.

$$-3x^{2}+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 0 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-3\right)\times 19}}{2\left(-3\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-3\right)\times 19}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{0±\sqrt{12\times 19}}{2\left(-3\right)}$$

Multiply 12 times 19.

$$x=\frac{0±\sqrt{228}}{2\left(-3\right)}$$

Take the square root of 228.

$$x=\frac{0±2\sqrt{57}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{0±2\sqrt{57}}{-6}$$

Now solve the equation x=\frac{0±2\sqrt{57}}{-6} when ± is plus.

$$x=-\frac{\sqrt{57}}{3}$$

Now solve the equation x=\frac{0±2\sqrt{57}}{-6} when ± is minus.

$$x=\frac{\sqrt{57}}{3}$$

The equation is now solved.

$$x=-\frac{\sqrt{57}}{3} x=\frac{\sqrt{57}}{3}$$