Solve 3x/x+2-x-1/6=x

$$\frac { 3 x } { x + 2 } - \frac { x - 1 } { 6 } = x$$

Solve for x

$$x=-\frac{2}{7}\approx -0.285714286$$

$$x=1$$

Steps Using Factoring By Grouping

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 6\left(x+2\right), the least common multiple of x+2,6.

$$6\times 3x-\left(x+2\right)\left(x-1\right)=x\times 6\left(x+2\right)$$

Multiply 6 and 3 to get 18.

$$18x-\left(x+2\right)\left(x-1\right)=x\times 6\left(x+2\right)$$

Use the distributive property to multiply x+2 by x-1 and combine like terms.

$$18x-\left(x^{2}+x-2\right)=x\times 6\left(x+2\right)$$

To find the opposite of x^{2}+x-2, find the opposite of each term.

$$18x-x^{2}-x+2=x\times 6\left(x+2\right)$$

Combine 18x and -x to get 17x.

$$17x-x^{2}+2=x\times 6\left(x+2\right)$$

Use the distributive property to multiply x\times 6 by x+2.

$$17x-x^{2}+2=6x^{2}+2x\times 6$$

Multiply 2 and 6 to get 12.

$$17x-x^{2}+2=6x^{2}+12x$$

Subtract 6x^{2} from both sides.

$$17x-x^{2}+2-6x^{2}=12x$$

Combine -x^{2} and -6x^{2} to get -7x^{2}.

$$17x-7x^{2}+2=12x$$

Subtract 12x from both sides.

$$17x-7x^{2}+2-12x=0$$

Combine 17x and -12x to get 5x.

$$5x-7x^{2}+2=0$$

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

$$-7x^{2}+5x+2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=5 ab=-7\times 2=-14$$

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 -14.

$$-1,14 -2,7$$

Calculate the sum for each pair.

$$-1+14=13 -2+7=5$$

The solution is the pair that gives sum 5.

$$a=7 b=-2$$

Rewrite -7x^{2}+5x+2 as \left(-7x^{2}+7x\right)+\left(-2x+2\right).

$$\left(-7x^{2}+7x\right)+\left(-2x+2\right)$$

Factor out 7x in the first and 2 in the second group.

$$7x\left(-x+1\right)+2\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(7x+2\right)$$

To find equation solutions, solve -x+1=0 and 7x+2=0.

$$x=1 x=-\frac{2}{7}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 6\left(x+2\right), the least common multiple of x+2,6.

$$6\times 3x-\left(x+2\right)\left(x-1\right)=x\times 6\left(x+2\right)$$

Multiply 6 and 3 to get 18.

$$18x-\left(x+2\right)\left(x-1\right)=x\times 6\left(x+2\right)$$

Use the distributive property to multiply x+2 by x-1 and combine like terms.

$$18x-\left(x^{2}+x-2\right)=x\times 6\left(x+2\right)$$

To find the opposite of x^{2}+x-2, find the opposite of each term.

$$18x-x^{2}-x+2=x\times 6\left(x+2\right)$$

Combine 18x and -x to get 17x.

$$17x-x^{2}+2=x\times 6\left(x+2\right)$$

Use the distributive property to multiply x\times 6 by x+2.

$$17x-x^{2}+2=6x^{2}+2x\times 6$$

Multiply 2 and 6 to get 12.

$$17x-x^{2}+2=6x^{2}+12x$$

Subtract 6x^{2} from both sides.

$$17x-x^{2}+2-6x^{2}=12x$$

Combine -x^{2} and -6x^{2} to get -7x^{2}.

$$17x-7x^{2}+2=12x$$

Subtract 12x from both sides.

$$17x-7x^{2}+2-12x=0$$

Combine 17x and -12x to get 5x.

$$5x-7x^{2}+2=0$$

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.

$$-7x^{2}+5x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 5 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-5±\sqrt{25+28\times 2}}{2\left(-7\right)}$$

Multiply 28 times 2.

$$x=\frac{-5±\sqrt{25+56}}{2\left(-7\right)}$$

Add 25 to 56.

$$x=\frac{-5±\sqrt{81}}{2\left(-7\right)}$$

Take the square root of 81.

$$x=\frac{-5±9}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-5±9}{-14}$$

Now solve the equation x=\frac{-5±9}{-14} when ± is plus. Add -5 to 9.

$$x=\frac{4}{-14}$$

Reduce the fraction \frac{4}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{7}$$

Now solve the equation x=\frac{-5±9}{-14} when ± is minus. Subtract 9 from -5.

$$x=-\frac{14}{-14}$$

Divide -14 by -14.

$$x=1$$

The equation is now solved.

$$x=-\frac{2}{7} x=1$$

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 6\left(x+2\right), the least common multiple of x+2,6.

$$6\times 3x-\left(x+2\right)\left(x-1\right)=x\times 6\left(x+2\right)$$

Multiply 6 and 3 to get 18.

$$18x-\left(x+2\right)\left(x-1\right)=x\times 6\left(x+2\right)$$

Use the distributive property to multiply x+2 by x-1 and combine like terms.

$$18x-\left(x^{2}+x-2\right)=x\times 6\left(x+2\right)$$

To find the opposite of x^{2}+x-2, find the opposite of each term.

$$18x-x^{2}-x+2=x\times 6\left(x+2\right)$$

Combine 18x and -x to get 17x.

$$17x-x^{2}+2=x\times 6\left(x+2\right)$$

Use the distributive property to multiply x\times 6 by x+2.

$$17x-x^{2}+2=6x^{2}+2x\times 6$$

Multiply 2 and 6 to get 12.

$$17x-x^{2}+2=6x^{2}+12x$$

Subtract 6x^{2} from both sides.

$$17x-x^{2}+2-6x^{2}=12x$$

Combine -x^{2} and -6x^{2} to get -7x^{2}.

$$17x-7x^{2}+2=12x$$

Subtract 12x from both sides.

$$17x-7x^{2}+2-12x=0$$

Combine 17x and -12x to get 5x.

$$5x-7x^{2}+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$5x-7x^{2}=-2$$

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.

$$-7x^{2}+5x=-2$$

Divide both sides by -7.

$$\frac{-7x^{2}+5x}{-7}=-\frac{2}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{5}{-7}x=-\frac{2}{-7}$$

Divide 5 by -7.

$$x^{2}-\frac{5}{7}x=-\frac{2}{-7}$$

Divide -2 by -7.

$$x^{2}-\frac{5}{7}x=\frac{2}{7}$$

Divide -\frac{5}{7}, the coefficient of the x term, by 2 to get -\frac{5}{14}. Then add the square of -\frac{5}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{7}x+\left(-\frac{5}{14}\right)^{2}=\frac{2}{7}+\left(-\frac{5}{14}\right)^{2}$$

Square -\frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=\frac{2}{7}+\frac{25}{196}$$

Add \frac{2}{7} to \frac{25}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{7}x+\frac{25}{196}=\frac{81}{196}$$

Factor x^{2}-\frac{5}{7}x+\frac{25}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{14}\right)^{2}=\frac{81}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{14}\right)^{2}}=\sqrt{\frac{81}{196}}$$

Simplify.

$$x-\frac{5}{14}=\frac{9}{14} x-\frac{5}{14}=-\frac{9}{14}$$

Add \frac{5}{14} to both sides of the equation.

$$x=1 x=-\frac{2}{7}$$