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

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

Solve for x

$$x=-6$$

$$x=5$$

Steps Using the Quadratic Formula

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

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

Multiply x+3 and x+3 to get \left(x+3\right)^{2}.

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

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

$$2x^{2}-8x+6+\left(x+3\right)^{2}=5\left(x-3\right)\left(x+3\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$2x^{2}-8x+6+x^{2}+6x+9=5\left(x-3\right)\left(x+3\right)$$

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

$$3x^{2}-8x+6+6x+9=5\left(x-3\right)\left(x+3\right)$$

Combine -8x and 6x to get -2x.

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

Add 6 and 9 to get 15.

$$3x^{2}-2x+15=5\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 5 by x-3.

$$3x^{2}-2x+15=\left(5x-15\right)\left(x+3\right)$$

Use the distributive property to multiply 5x-15 by x+3 and combine like terms.

$$3x^{2}-2x+15=5x^{2}-45$$

Subtract 5x^{2} from both sides.

$$3x^{2}-2x+15-5x^{2}=-45$$

Combine 3x^{2} and -5x^{2} to get -2x^{2}.

$$-2x^{2}-2x+15=-45$$

Add 45 to both sides.

$$-2x^{2}-2x+15+45=0$$

Add 15 and 45 to get 60.

$$-2x^{2}-2x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -2 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-2\right)\times 60}}{2\left(-2\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-2\right)\times 60}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-2\right)±\sqrt{4+8\times 60}}{2\left(-2\right)}$$

Multiply 8 times 60.

$$x=\frac{-\left(-2\right)±\sqrt{4+480}}{2\left(-2\right)}$$

Add 4 to 480.

$$x=\frac{-\left(-2\right)±\sqrt{484}}{2\left(-2\right)}$$

Take the square root of 484.

$$x=\frac{-\left(-2\right)±22}{2\left(-2\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±22}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{2±22}{-4}$$

Now solve the equation x=\frac{2±22}{-4} when ± is plus. Add 2 to 22.

$$x=\frac{24}{-4}$$

Divide 24 by -4.

$$x=-6$$

Now solve the equation x=\frac{2±22}{-4} when ± is minus. Subtract 22 from 2.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

The equation is now solved.

$$x=-6 x=5$$

Steps for Completing the Square

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

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

Multiply x+3 and x+3 to get \left(x+3\right)^{2}.

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

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

$$2x^{2}-8x+6+\left(x+3\right)^{2}=5\left(x-3\right)\left(x+3\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$2x^{2}-8x+6+x^{2}+6x+9=5\left(x-3\right)\left(x+3\right)$$

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

$$3x^{2}-8x+6+6x+9=5\left(x-3\right)\left(x+3\right)$$

Combine -8x and 6x to get -2x.

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

Add 6 and 9 to get 15.

$$3x^{2}-2x+15=5\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 5 by x-3.

$$3x^{2}-2x+15=\left(5x-15\right)\left(x+3\right)$$

Use the distributive property to multiply 5x-15 by x+3 and combine like terms.

$$3x^{2}-2x+15=5x^{2}-45$$

Subtract 5x^{2} from both sides.

$$3x^{2}-2x+15-5x^{2}=-45$$

Combine 3x^{2} and -5x^{2} to get -2x^{2}.

$$-2x^{2}-2x+15=-45$$

Subtract 15 from both sides.

$$-2x^{2}-2x=-45-15$$

Subtract 15 from -45 to get -60.

$$-2x^{2}-2x=-60$$

Divide both sides by -2.

$$\frac{-2x^{2}-2x}{-2}=-\frac{60}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{2}{-2}\right)x=-\frac{60}{-2}$$

Divide -2 by -2.

$$x^{2}+x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x^{2}+x=30$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=30+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=30+\frac{1}{4}$$

Add 30 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{121}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{11}{2} x+\frac{1}{2}=-\frac{11}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=5 x=-6$$