Solve x+2/x+3+x-3/x-2=5/6

$$\frac { x + 2 } { x + 3 } + \frac { x - 3 } { x - 2 } = \frac { 5 } { 6 }$$

Solve for x

$$x = -\frac{16}{7} = -2\frac{2}{7} \approx -2.285714286$$

$$x=3$$

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 6\left(x-2\right)\left(x+3\right), the least common multiple of x+3,x-2,6.

$$\left(6x-12\right)\left(x+2\right)+\left(6x+18\right)\left(x-3\right)=5\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-12 by x+2 and combine like terms.

$$6x^{2}-24+\left(6x+18\right)\left(x-3\right)=5\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 6x+18 by x-3 and combine like terms.

$$6x^{2}-24+6x^{2}-54=5\left(x-2\right)\left(x+3\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}-24-54=5\left(x-2\right)\left(x+3\right)$$

Subtract 54 from -24 to get -78.

$$12x^{2}-78=5\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 5 by x-2.

$$12x^{2}-78=\left(5x-10\right)\left(x+3\right)$$

Use the distributive property to multiply 5x-10 by x+3 and combine like terms.

$$12x^{2}-78=5x^{2}+5x-30$$

Subtract 5x^{2} from both sides.

$$12x^{2}-78-5x^{2}=5x-30$$

Combine 12x^{2} and -5x^{2} to get 7x^{2}.

$$7x^{2}-78=5x-30$$

Subtract 5x from both sides.

$$7x^{2}-78-5x=-30$$

Add 30 to both sides.

$$7x^{2}-78-5x+30=0$$

Add -78 and 30 to get -48.

$$7x^{2}-48-5x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$7x^{2}-5x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -5 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 7\left(-48\right)}}{2\times 7}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 7\left(-48\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-5\right)±\sqrt{25-28\left(-48\right)}}{2\times 7}$$

Multiply -28 times -48.

$$x=\frac{-\left(-5\right)±\sqrt{25+1344}}{2\times 7}$$

Add 25 to 1344.

$$x=\frac{-\left(-5\right)±\sqrt{1369}}{2\times 7}$$

Take the square root of 1369.

$$x=\frac{-\left(-5\right)±37}{2\times 7}$$

The opposite of -5 is 5.

$$x=\frac{5±37}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{5±37}{14}$$

Now solve the equation x=\frac{5±37}{14} when ± is plus. Add 5 to 37.

$$x=\frac{42}{14}$$

Divide 42 by 14.

$$x=3$$

Now solve the equation x=\frac{5±37}{14} when ± is minus. Subtract 37 from 5.

$$x=-\frac{32}{14}$$

Reduce the fraction \frac{-32}{14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{16}{7}$$

The equation is now solved.

$$x=3 x=-\frac{16}{7}$$

Steps for Completing the Square

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 6\left(x-2\right)\left(x+3\right), the least common multiple of x+3,x-2,6.

$$\left(6x-12\right)\left(x+2\right)+\left(6x+18\right)\left(x-3\right)=5\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-12 by x+2 and combine like terms.

$$6x^{2}-24+\left(6x+18\right)\left(x-3\right)=5\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 6x+18 by x-3 and combine like terms.

$$6x^{2}-24+6x^{2}-54=5\left(x-2\right)\left(x+3\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}-24-54=5\left(x-2\right)\left(x+3\right)$$

Subtract 54 from -24 to get -78.

$$12x^{2}-78=5\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 5 by x-2.

$$12x^{2}-78=\left(5x-10\right)\left(x+3\right)$$

Use the distributive property to multiply 5x-10 by x+3 and combine like terms.

$$12x^{2}-78=5x^{2}+5x-30$$

Subtract 5x^{2} from both sides.

$$12x^{2}-78-5x^{2}=5x-30$$

Combine 12x^{2} and -5x^{2} to get 7x^{2}.

$$7x^{2}-78=5x-30$$

Subtract 5x from both sides.

$$7x^{2}-78-5x=-30$$

Add 78 to both sides.

$$7x^{2}-5x=-30+78$$

Add -30 and 78 to get 48.

$$7x^{2}-5x=48$$

Divide both sides by 7.

$$\frac{7x^{2}-5x}{7}=\frac{48}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{5}{7}x=\frac{48}{7}$$

Divide -\frac{5}{7}, the coefficient of the x term, by 2 to get -\frac{5}{14}. Then add the square of -\frac{5}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{7}x+\left(-\frac{5}{14}\right)^{2}=\frac{48}{7}+\left(-\frac{5}{14}\right)^{2}$$

Square -\frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=\frac{48}{7}+\frac{25}{196}$$

Add \frac{48}{7} to \frac{25}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=\frac{1369}{196}$$

Factor x^{2}-\frac{5}{7}x+\frac{25}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{14}\right)^{2}=\frac{1369}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{14}\right)^{2}}=\sqrt{\frac{1369}{196}}$$

Simplify.

$$x-\frac{5}{14}=\frac{37}{14} x-\frac{5}{14}=-\frac{37}{14}$$

Add \frac{5}{14} to both sides of the equation.

$$x=3 x=-\frac{16}{7}$$