Solve 1/x+10-1/x=1/12

$$\frac { 1 } { x + 10 } - \frac { 1 } { x } = \frac { 1 } { 12 }$$

Solve for x (complex solution)

$$x=-5+\sqrt{95}i\approx -5+9.746794345i$$

$$x=-\sqrt{95}i-5\approx -5-9.746794345i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+10\right), the least common multiple of x+10,x,12.

$$12x-\left(12x+120\right)=x\left(x+10\right)$$

To find the opposite of 12x+120, find the opposite of each term.

$$12x-12x-120=x\left(x+10\right)$$

Combine 12x and -12x to get 0.

$$-120=x\left(x+10\right)$$

Use the distributive property to multiply x by x+10.

$$-120=x^{2}+10x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+10x=-120$$

Add 120 to both sides.

$$x^{2}+10x+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 120}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 120}}{2}$$

Multiply -4 times 120.

$$x=\frac{-10±\sqrt{100-480}}{2}$$

Add 100 to -480.

$$x=\frac{-10±\sqrt{-380}}{2}$$

Take the square root of -380.

$$x=\frac{-10±2\sqrt{95}i}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{95}i}{2} when ± is plus. Add -10 to 2i\sqrt{95}.

$$x=\frac{-10+2\sqrt{95}i}{2}$$

Divide -10+2i\sqrt{95} by 2.

$$x=-5+\sqrt{95}i$$

Now solve the equation x=\frac{-10±2\sqrt{95}i}{2} when ± is minus. Subtract 2i\sqrt{95} from -10.

$$x=\frac{-2\sqrt{95}i-10}{2}$$

Divide -10-2i\sqrt{95} by 2.

$$x=-\sqrt{95}i-5$$

The equation is now solved.

$$x=-5+\sqrt{95}i x=-\sqrt{95}i-5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+10\right), the least common multiple of x+10,x,12.

$$12x-\left(12x+120\right)=x\left(x+10\right)$$

To find the opposite of 12x+120, find the opposite of each term.

$$12x-12x-120=x\left(x+10\right)$$

Combine 12x and -12x to get 0.

$$-120=x\left(x+10\right)$$

Use the distributive property to multiply x by x+10.

$$-120=x^{2}+10x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+10x=-120$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=-120+5^{2}$$

Square 5.

$$x^{2}+10x+25=-120+25$$

Add -120 to 25.

$$x^{2}+10x+25=-95$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=-95$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{-95}$$

Simplify.

$$x+5=\sqrt{95}i x+5=-\sqrt{95}i$$

Subtract 5 from both sides of the equation.

$$x=-5+\sqrt{95}i x=-\sqrt{95}i-5$$