Solve 1/x+1/x+4=1/3

$$\frac { 1 } { x } + \frac { 1 } { x + 4 } = \frac { 1 } { 3 }$$

Solve for x

$$x=\sqrt{13}+1\approx 4.605551275$$

$$x=1-\sqrt{13}\approx -2.605551275$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+4\right), the least common multiple of x,x+4,3.

$$3x+12+3x=x\left(x+4\right)$$

Combine 3x and 3x to get 6x.

$$6x+12=x\left(x+4\right)$$

Use the distributive property to multiply x by x+4.

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

Subtract x^{2} from both sides.

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

Subtract 4x from both sides.

$$6x+12-x^{2}-4x=0$$

Combine 6x and -4x to get 2x.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$x=\frac{-2±\sqrt{4+48}}{2\left(-1\right)}$$

Add 4 to 48.

$$x=\frac{-2±\sqrt{52}}{2\left(-1\right)}$$

Take the square root of 52.

$$x=\frac{-2±2\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±2\sqrt{13}}{-2}$$

Now solve the equation x=\frac{-2±2\sqrt{13}}{-2} when ± is plus. Add -2 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}-2}{-2}$$

Divide -2+2\sqrt{13} by -2.

$$x=1-\sqrt{13}$$

Now solve the equation x=\frac{-2±2\sqrt{13}}{-2} when ± is minus. Subtract 2\sqrt{13} from -2.

$$x=\frac{-2\sqrt{13}-2}{-2}$$

Divide -2-2\sqrt{13} by -2.

$$x=\sqrt{13}+1$$

The equation is now solved.

$$x=1-\sqrt{13} x=\sqrt{13}+1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+4\right), the least common multiple of x,x+4,3.

$$3x+12+3x=x\left(x+4\right)$$

Combine 3x and 3x to get 6x.

$$6x+12=x\left(x+4\right)$$

Use the distributive property to multiply x by x+4.

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

Subtract x^{2} from both sides.

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

Subtract 4x from both sides.

$$6x+12-x^{2}-4x=0$$

Combine 6x and -4x to get 2x.

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

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=-\frac{12}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=-\frac{12}{-1}$$

Divide -12 by -1.

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

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

$$x^{2}-2x+1=12+1$$

Add 12 to 1.

$$x^{2}-2x+1=13$$

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

$$\left(x-1\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{13}$$

Simplify.

$$x-1=\sqrt{13} x-1=-\sqrt{13}$$

Add 1 to both sides of the equation.

$$x=\sqrt{13}+1 x=1-\sqrt{13}$$