Solve 5/x-5/x-1=x+6/2x-2

$$\frac{ 5 }{ x } - \frac{ 5 }{ x-1 } = \frac{ x+6 }{ 2x-2 }$$

Solve for x (complex solution)

$$x=-3-i$$

$$x=-3+i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-1\right), the least common multiple of x,x-1,2x-2.

$$\left(2x-2\right)\times 5-2x\times 5=x\left(x+6\right)$$

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

$$10x-10-2x\times 5=x\left(x+6\right)$$

Multiply 2 and 5 to get 10.

$$10x-10-10x=x\left(x+6\right)$$

Use the distributive property to multiply x by x+6.

$$10x-10-10x=x^{2}+6x$$

Subtract x^{2} from both sides.

$$10x-10-10x-x^{2}=6x$$

Subtract 6x from both sides.

$$10x-10-10x-x^{2}-6x=0$$

Combine 10x and -6x to get 4x.

$$4x-10-10x-x^{2}=0$$

Combine 4x and -10x to get -6x.

$$-6x-10-x^{2}=0$$

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.

$$-x^{2}-6x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\left(-10\right)}}{2\left(-1\right)}$$

Multiply 4 times -10.

$$x=\frac{-\left(-6\right)±\sqrt{36-40}}{2\left(-1\right)}$$

Add 36 to -40.

$$x=\frac{-\left(-6\right)±\sqrt{-4}}{2\left(-1\right)}$$

Take the square root of -4.

$$x=\frac{-\left(-6\right)±2i}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±2i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±2i}{-2}$$

Now solve the equation x=\frac{6±2i}{-2} when ± is plus. Add 6 to 2i.

$$x=\frac{6+2i}{-2}$$

Divide 6+2i by -2.

$$x=-3-i$$

Now solve the equation x=\frac{6±2i}{-2} when ± is minus. Subtract 2i from 6.

$$x=\frac{6-2i}{-2}$$

Divide 6-2i by -2.

$$x=-3+i$$

The equation is now solved.

$$x=-3-i x=-3+i$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-1\right), the least common multiple of x,x-1,2x-2.

$$\left(2x-2\right)\times 5-2x\times 5=x\left(x+6\right)$$

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

$$10x-10-2x\times 5=x\left(x+6\right)$$

Multiply 2 and 5 to get 10.

$$10x-10-10x=x\left(x+6\right)$$

Use the distributive property to multiply x by x+6.

$$10x-10-10x=x^{2}+6x$$

Subtract x^{2} from both sides.

$$10x-10-10x-x^{2}=6x$$

Subtract 6x from both sides.

$$10x-10-10x-x^{2}-6x=0$$

Combine 10x and -6x to get 4x.

$$4x-10-10x-x^{2}=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$4x-10x-x^{2}=10$$

Combine 4x and -10x to get -6x.

$$-6x-x^{2}=10$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}-6x=10$$

Divide both sides by -1.

$$\frac{-x^{2}-6x}{-1}=\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{6}{-1}\right)x=\frac{10}{-1}$$

Divide -6 by -1.

$$x^{2}+6x=\frac{10}{-1}$$

Divide 10 by -1.

$$x^{2}+6x=-10$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=-10+3^{2}$$

Square 3.

$$x^{2}+6x+9=-10+9$$

Add -10 to 9.

$$x^{2}+6x+9=-1$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+3=i x+3=-i$$

Subtract 3 from both sides of the equation.

$$x=-3+i x=-3-i$$