Solve 1/x+1/x+6-1/4=0

$$\frac { 1 } { x } + \frac { 1 } { x + 6 } - \frac { 1 } { 4 } = 0$$

Solve for x

$$x=-4$$

$$x=6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+6\right), the least common multiple of x,x+6,4.

$$4x+24+4x+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Combine 4x and 4x to get 8x.

$$8x+24+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Multiply 4 and -\frac{1}{4} to get -1.

$$8x+24-x\left(x+6\right)=0$$

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

$$8x+24-x^{2}-6x=0$$

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

$$2x+24-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+2x+24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=2 ab=-24=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 2.

$$a=6 b=-4$$

Rewrite -x^{2}+2x+24 as \left(-x^{2}+6x\right)+\left(-4x+24\right).

$$\left(-x^{2}+6x\right)+\left(-4x+24\right)$$

Factor out -x in the first and -4 in the second group.

$$-x\left(x-6\right)-4\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x-4\right)$$

To find equation solutions, solve x-6=0 and -x-4=0.

$$x=6 x=-4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+6\right), the least common multiple of x,x+6,4.

$$4x+24+4x+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Combine 4x and 4x to get 8x.

$$8x+24+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Multiply 4 and -\frac{1}{4} to get -1.

$$8x+24-x\left(x+6\right)=0$$

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

$$8x+24-x^{2}-6x=0$$

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

$$2x+24-x^{2}=0$$

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.

$$-x^{2}+2x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$x=\frac{-2±\sqrt{4+96}}{2\left(-1\right)}$$

Add 4 to 96.

$$x=\frac{-2±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-2±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±10}{-2}$$

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

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

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

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

Divide -12 by -2.

$$x=6$$

The equation is now solved.

$$x=-4 x=6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+6\right), the least common multiple of x,x+6,4.

$$4x+24+4x+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Combine 4x and 4x to get 8x.

$$8x+24+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Multiply 4 and -\frac{1}{4} to get -1.

$$8x+24-x\left(x+6\right)=0$$

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

$$8x+24-x^{2}-6x=0$$

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

$$2x+24-x^{2}=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$2x-x^{2}=-24$$

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.

$$-x^{2}+2x=-24$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=-\frac{24}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=-\frac{24}{-1}$$

Divide -24 by -1.

$$x^{2}-2x=24$$

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.

$$x^{2}-2x+1=24+1$$

Add 24 to 1.

$$x^{2}-2x+1=25$$

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}.

$$\left(x-1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-1=5 x-1=-5$$

Add 1 to both sides of the equation.

$$x=6 x=-4$$