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

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

Solve for x

$$x=-2$$

$$x=3$$

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 x\left(x+6\right), the least common multiple of x+6,x.

$$x\left(x+1\right)+x+6=x\left(2x+1\right)$$

Use the distributive property to multiply x by x+1.

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

Combine x and x to get 2x.

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

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

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

Subtract 2x^{2} from both sides.

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

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

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

Subtract x from both sides.

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

Combine 2x and -x to get x.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=1 ab=-6=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 1.

$$a=3 b=-2$$

Rewrite -x^{2}+x+6 as \left(-x^{2}+3x\right)+\left(-2x+6\right).

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

Factor out -x in the first and -2 in the second group.

$$-x\left(x-3\right)-2\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(-x-2\right)$$

To find equation solutions, solve x-3=0 and -x-2=0.

$$x=3 x=-2$$

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 x\left(x+6\right), the least common multiple of x+6,x.

$$x\left(x+1\right)+x+6=x\left(2x+1\right)$$

Use the distributive property to multiply x by x+1.

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

Combine x and x to get 2x.

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

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

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

Subtract 2x^{2} from both sides.

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

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

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

Subtract x from both sides.

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

Combine 2x and -x to get x.

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

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

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times 6.

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

Add 1 to 24.

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

Take the square root of 25.

$$x=\frac{-1±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±5}{-2}$$

Now solve the equation x=\frac{-1±5}{-2} when ± is plus. Add -1 to 5.

$$x=\frac{4}{-2}$$

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-1±5}{-2} when ± is minus. Subtract 5 from -1.

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

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=-2 x=3$$

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 x\left(x+6\right), the least common multiple of x+6,x.

$$x\left(x+1\right)+x+6=x\left(2x+1\right)$$

Use the distributive property to multiply x by x+1.

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

Combine x and x to get 2x.

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

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

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

Subtract 2x^{2} from both sides.

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

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

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

Subtract x from both sides.

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

Combine 2x and -x to get x.

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

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide -6 by -1.

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

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

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=6+\frac{1}{4}$$

Add 6 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{25}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

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

Add \frac{1}{2} to both sides of the equation.

$$x=3 x=-2$$