Solve 3/x-2-2/x+3=1

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

Solve for x

$$x=\sqrt{19}\approx 4.358898944$$

$$x=-\sqrt{19}\approx -4.358898944$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -3,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+3\right), the least common multiple of x-2,x+3.

$$\left(x+3\right)\times 3-\left(x-2\right)\times 2=\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 3.

$$3x+9-\left(x-2\right)\times 2=\left(x-2\right)\left(x+3\right)$$

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

$$3x+9-\left(2x-4\right)=\left(x-2\right)\left(x+3\right)$$

To find the opposite of 2x-4, find the opposite of each term.

$$3x+9-2x+4=\left(x-2\right)\left(x+3\right)$$

Combine 3x and -2x to get x.

$$x+9+4=\left(x-2\right)\left(x+3\right)$$

Add 9 and 4 to get 13.

$$x+13=\left(x-2\right)\left(x+3\right)$$

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

$$x+13=x^{2}+x-6$$

Subtract x^{2} from both sides.

$$x+13-x^{2}=x-6$$

Subtract x from both sides.

$$x+13-x^{2}-x=-6$$

Combine x and -x to get 0.

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

Subtract 13 from both sides.

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

Subtract 13 from -6 to get -19.

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

Divide both sides by -1.

$$x^{2}=\frac{-19}{-1}$$

Fraction \frac{-19}{-1} can be simplified to 19 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=19$$

Take the square root of both sides of the equation.

$$x=\sqrt{19} x=-\sqrt{19}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+3\right), the least common multiple of x-2,x+3.

$$\left(x+3\right)\times 3-\left(x-2\right)\times 2=\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 3.

$$3x+9-\left(x-2\right)\times 2=\left(x-2\right)\left(x+3\right)$$

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

$$3x+9-\left(2x-4\right)=\left(x-2\right)\left(x+3\right)$$

To find the opposite of 2x-4, find the opposite of each term.

$$3x+9-2x+4=\left(x-2\right)\left(x+3\right)$$

Combine 3x and -2x to get x.

$$x+9+4=\left(x-2\right)\left(x+3\right)$$

Add 9 and 4 to get 13.

$$x+13=\left(x-2\right)\left(x+3\right)$$

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

$$x+13=x^{2}+x-6$$

Subtract x^{2} from both sides.

$$x+13-x^{2}=x-6$$

Subtract x from both sides.

$$x+13-x^{2}-x=-6$$

Combine x and -x to get 0.

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

Add 6 to both sides.

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

Add 13 and 6 to get 19.

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

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 19}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\times 19}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\times 19}}{2\left(-1\right)}$$

Multiply 4 times 19.

$$x=\frac{0±\sqrt{76}}{2\left(-1\right)}$$

Take the square root of 76.

$$x=\frac{0±2\sqrt{19}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±2\sqrt{19}}{-2}$$

Now solve the equation x=\frac{0±2\sqrt{19}}{-2} when ± is plus.

$$x=-\sqrt{19}$$

Now solve the equation x=\frac{0±2\sqrt{19}}{-2} when ± is minus.

$$x=\sqrt{19}$$

The equation is now solved.

$$x=-\sqrt{19} x=\sqrt{19}$$