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

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

Solve for x (complex solution)

$$x=\frac{7+\sqrt{23}i}{2}\approx 3.5+2.397915762i$$

$$x=\frac{-\sqrt{23}i+7}{2}\approx 3.5-2.397915762i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+3\right), the least common multiple of x+3,2x-2.

$$\left(2x-2\right)\left(x-3\right)+\left(x+3\right)\left(x+2\right)=2\left(x-1\right)\left(x+3\right)$$

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

$$2x^{2}-8x+6+\left(x+3\right)\left(x+2\right)=2\left(x-1\right)\left(x+3\right)$$

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

$$2x^{2}-8x+6+x^{2}+5x+6=2\left(x-1\right)\left(x+3\right)$$

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

$$3x^{2}-8x+6+5x+6=2\left(x-1\right)\left(x+3\right)$$

Combine -8x and 5x to get -3x.

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

Add 6 and 6 to get 12.

$$3x^{2}-3x+12=2\left(x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 2 by x-1.

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

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

$$3x^{2}-3x+12=2x^{2}+4x-6$$

Subtract 2x^{2} from both sides.

$$3x^{2}-3x+12-2x^{2}=4x-6$$

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

$$x^{2}-3x+12=4x-6$$

Subtract 4x from both sides.

$$x^{2}-3x+12-4x=-6$$

Combine -3x and -4x to get -7x.

$$x^{2}-7x+12=-6$$

Add 6 to both sides.

$$x^{2}-7x+12+6=0$$

Add 12 and 6 to get 18.

$$x^{2}-7x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 18}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-\left(-7\right)±\sqrt{49-72}}{2}$$

Add 49 to -72.

$$x=\frac{-\left(-7\right)±\sqrt{-23}}{2}$$

Take the square root of -23.

$$x=\frac{-\left(-7\right)±\sqrt{23}i}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{23}i}{2}$$

Now solve the equation x=\frac{7±\sqrt{23}i}{2} when ± is plus. Add 7 to i\sqrt{23}.

$$x=\frac{7+\sqrt{23}i}{2}$$

Now solve the equation x=\frac{7±\sqrt{23}i}{2} when ± is minus. Subtract i\sqrt{23} from 7.

$$x=\frac{-\sqrt{23}i+7}{2}$$

The equation is now solved.

$$x=\frac{7+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+7}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+3\right), the least common multiple of x+3,2x-2.

$$\left(2x-2\right)\left(x-3\right)+\left(x+3\right)\left(x+2\right)=2\left(x-1\right)\left(x+3\right)$$

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

$$2x^{2}-8x+6+\left(x+3\right)\left(x+2\right)=2\left(x-1\right)\left(x+3\right)$$

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

$$2x^{2}-8x+6+x^{2}+5x+6=2\left(x-1\right)\left(x+3\right)$$

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

$$3x^{2}-8x+6+5x+6=2\left(x-1\right)\left(x+3\right)$$

Combine -8x and 5x to get -3x.

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

Add 6 and 6 to get 12.

$$3x^{2}-3x+12=2\left(x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 2 by x-1.

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

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

$$3x^{2}-3x+12=2x^{2}+4x-6$$

Subtract 2x^{2} from both sides.

$$3x^{2}-3x+12-2x^{2}=4x-6$$

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

$$x^{2}-3x+12=4x-6$$

Subtract 4x from both sides.

$$x^{2}-3x+12-4x=-6$$

Combine -3x and -4x to get -7x.

$$x^{2}-7x+12=-6$$

Subtract 12 from both sides.

$$x^{2}-7x=-6-12$$

Subtract 12 from -6 to get -18.

$$x^{2}-7x=-18$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-18+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-18+\frac{49}{4}$$

Add -18 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=-\frac{23}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=-\frac{23}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{23}i}{2} x-\frac{7}{2}=-\frac{\sqrt{23}i}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{7+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+7}{2}$$